Although I can't provide a verdict as to whether forms of mass media led to the public distrusting information per se, I can talk about this question in relation to twentieth century British politics in a slightly different but still relevant manner: The way in which new forms of mass media changed the relationship between British politicians and their electorate. In this sense, I may be able to provide you with some context about the ways in which our perception of politicians has changed over the last century or so, which is closely related to how we trust the information they provide. I am wary of the "No Current Events" rule in this Subreddit, and so my answer will not encroach on the last twenty years. I must stress that this is a massive and very wide topic and without a doubt there are conclusions entirely different to the ones I've drawn. I'd love to hear them. This could get very political, which I'll do my best to avoid.

So let's look at how mass media changed relationships between politicians and the public.

Radio:

A.J.P. Taylor argued that innovations in mass media in the inter-war period fundamentally changed politics in Britain, into a "gentler, more intimate style". However, he attributes far too much importance to the role of the media. Lawrence has shown that the populism of the Edwardian period was already on the decline regardless because of the impact of the "War, Women and the 'Silent Majority'" (Lawrence, p.96-129):

• The First World War: A move towards more orderly politics was followed due to assumptions that citizens wished for more peaceable politicians.

• Female enfranchisement (1918): A new and substantial proportion of the electorate was created, who it was assumed sought rational arguments in their political inclinations. The Fourth Reform Act also limited election expenses and gave candidates schoolrooms to canvas, changing electioneering methods without the involvement of mass media.

• Labour's breakthrough in elections: Changes in voting suggested that there were plenty of uncommitted voters who could easily be won over. It also led to arguments that Labour was unfit for government because the party was unruly, and hence Labour's politicians made efforts to present themselves as more disciplined.

To sum up his argument, these influences led to the main parties moving towards rational arguments instead of vulgar electioneering and rowdyism (also see Rosenbaum). Efforts became focused on winning over less partisan citizens, most especially through effective and convincing arguments. Yes, radio in particular had an impact on this, but politicians also looked to the written word to convince the uncommitted voter. Lawrence quotes Morris in 1921, who claimed that "the old street corner rowdyism, higher coloured motor cars, and carefully staged torchlight processions were things of the past" (Lawrence, p.124). This was before radio and cinema became major influences in politics.

My point here is that it is often appropriate to look beyond technological determinism when trying to explain changes in political electioneering.

Nonetheless, Taylor is right to claim that mass media did contribute towards a more "intimate" style of politics, which had the potential for the public to place more trust in politicians. It was not the sole factor, but there are obvious examples which are relevant to your question.

Perhaps the most obvious example was the innovation of Baldwin in the "fireside chat" (see Nicholas, p.132-137). In radio broadcasts, Baldwin's political opponents chose to simply use their live speeches and put them over the air. These were horribly suited to radio. Baldwin, in contrast, spoke in a pre-recorded, friendly manner, as if he were speaking to the individual listening at home. He tactically had his wife knitting in the background, relied on personal anecdotes, and even paused every so often to light his pipe. This was very much a personal innovation almost unique to Baldwin at the time, but it was hugely effective and certainly led to people actually trusting him more. Lawrence provides some excellent examples of this, such as a man from Dagenham who wrote: "although I have never met you I feel I know you quite well ... I feel I can trust you ... you typify in yourself that innate steadiness and probity that is Britain". Another claimed that "we do know you Sir, and we do trust you" (Lawrence, p.97).

Cinema:

Baldwin's innovation also translated well to the use of cinema and newsreels. I think it's really important to appreciate that the use of radio and cinema allowed voters to become genuinely familiar with politicians, in a way the vast majority had never before found possible. Baldwin exploited this hugely effectively by portraying himself as a countryman, an Englishman; instead of creating just an image, he sought to cultivate a set of political values (Ramsden, p.200-214). This only enhanced the public's trust vested in him. To quote Lawrence again, on his election broadcast in 1935:

According to one correspondent, his [Baldwin's] address was the only one to be applauded in her local Wood Green cinema, where a working man was overheard saying "What I likes about Baldwin, 'e don't sling no mud" (Lawrence, p.99).

Cinema provided another means in which politicians could form new bonds with a wider section of the electorate that ever before, and relate to them on a personal level.

Newspapers:

It's easy to exaggerate the importance of both radio and cinema in the inter-war period. In 1925, only one in five homes owned a radio. By 1930, ownership was only at a third of homes. And in terms of cinema, following the explicit Government propaganda used during the wars, overt politicisation of film was unpopular in Britain. We must remember that people still received political news, for the most part, by reading newspapers. This was hardly a new innovation, but during the twentieth century the importance of the press and its perceived reliability changed dramatically. Lawrence has used Mass Observation reports to show this, which concluded in 1931 that:

"people regard newspapers as a source of the day's topics of conversation, and not as a source of truth about events. Similarly, a 1939 inquiry into 'broadcasting in everyday life' found that over 95 per cent of respondents "gave more credence to the broadcast News" than to what they read in the press" (Lawrence, p.100).

Part of the reason for this was that in the Edwardian period, the provincial press in each constituency often had separate newspapers for Liberals and Conservatives. These politically partisan newspapers could appeal to particular voters who subscribed to similar views, and provide what they saw as "reliable" information. As the twentieth century progressed, one single local paper dominated. Lawrence has claimed that this led to local papers becoming more politically neutral in order to maximise their readership (Lawrence, p.101), but this is somewhat misleading. Papers responded to popular demand more than political leanings, which meant their reporting became less reliable in the eyes of the public. On a national level, the obvious press interventions in politics also acted to reduce people's trust in them. Perhaps the most obvious example is the campaign of the "press barons" Lords Rothermere and Beaverbrook, who opposed both Lloyd George and Baldwin at certain times to such an extent that it served to discredit the press.

Radio and cinema, as mass media innovations, therefore served to actually restore trust in information, as an alternative to a press seen as unreliable.

Word limit reached - to split it properly I've put the next half as a reply :)

I also can't speak to the distrust of information, though there's a very long history of the effects of changes in media landscapes on social cultures.

The landmark work is Elisabeth Eisenstein's The Printing Press as an Agent of Change, which largely looks at the impact of printing itself on Europe from the 15th century onward, including the Schism of the Catholic Church, the 30 Years War, and the Renaissance. Eisenstein writes of how the printing press changed the relationship to the written, now printed, word, and the standarisation of texts.

I've been looking in particular at how the economies of information dissemination have changed. As of ~1400, there were roughly 30,000 books in Western Europe. Note, that's not titles, but books, in total. By 1500, the number was 200 million. (Buringh, Eltjo; van Zanden, Jan Luiten: "Charting the “Rise of the West”: Manuscripts and Printed Books in Europe, A Long-Term Perspective from the Sixth through Eighteenth Centuries", The Journal of Economic History, Vol. 69, No. 2 (2009), pp. 409–445 ).

The time to produce books fell tremendously -- a single scribe could copy from one to three books per year (various systems including farming out the copying to multiple scribes could reduce the time, but not labour, per book). The equivalent cost was on the order of $500k - $1m per book (largely based on labour equivalents -- precise pricing comparisons are difficult).

By 1776, when Adam Smith's Wealth of Nations was published, it cost 5 pounds, at a time when a workman's wages were roughly 20 pounds. Per year.

John R. Gillis, The Development of European Society, 1770-1870 discusses the rise in literacy (from ~25% to 90% over the 19th century), the growth in newspapers (as well as continuation of pamphlets), falling costs of paper, and the tremendous increase in the speed of printing presses (from ~180 impressions/hr to 10s of thousands, and finally 1 million or more by the end of the century) as facilitating the growth of populist movements.

One frequent observation is the growing power of mass media as a means of promoting disruptive political movements. The effect of Radio in Nazi Germany , as well as the other tools of mass media: audio capture and recording equipment (including tape recordings), radio, cinema, phonographs, ever cheaper and faster printing capabilities, photo-reproduction, and more.

https://www.theguardian.com/culture/2008/oct/09/radio.hitler.bbc.czechoslovakia

As to the fundamental question of the believability of speech and communications, that question goes back to Plato vs. the Sophists:

The philosophical problem of the nature of sophistry is arguably even more formidable. Due in large part to the influence of Plato and Aristotle, the term sophistry has come to signify the deliberate use of fallacious reasoning, intellectual charlatanism and moral unscrupulousness.

The practices of rhetoric vs. dialectics address this, as does the fundamental problem of conveying complex ideas to a large electorate. H.L. Mencken makes this the heart of his 1920 essay, "Brayard vs. Lionheart":

The first and last aim of the politician is to get votes, and the safest of all ways to get votes is to appear to the plain man to be a plain man like himself, which is to say, to appear to him to be happily free from any heretical treason to the body of accepted platitudes-to be filled to the brim with the flabby, banal, childish notions that challenge no prejudice and lay no burden of examination upon the mind.... The plain people ... shy instantly and inevitably from the man who comes before them with notions that they cannot immediately translate into terms of their everyday delusions; they fear the novel idea, and particularly the revolutionary idea, as they fear the devil.

Walter Lippmann's [Public Opinion]( https://en.m.wikipedia.org/wiki/Public_Opinion_(book)), considered a foundation document of modern journalism, lays out more of this dynamic, as does what's called the Lippmann-Dewey debate, contrasting two views of modern journalism.

Or, TL;DR: changing how people communicate, transmit, and receive information has deeply profound impacts on what is transmitted and how society itself as a whole responds.