This is a big, complicated question. Forgive me if I unpack it a bit.

For one, the "we" has got to be specified for any kind of answer. The question of how people determined what was true or accurate in a small farming community along the Congo River in Central Africa in 1750 is radically different than it was in a London coffeehouse in 1750. (As it is today, though I think the difference is less extensive today than it was then.) I think the OP is likely thinking of the "we" of urban North America and Western Europe in the 1700s and early 1800s, so let's go with that.

For another, this question is going to be hard to answer if we're trying to contrast our present behavior with the past in a way that is an unrealistic portrayal of the present. (Yes, I'm fact-checking a question about fact-checking.) Leaving aside all the current attention to "fake news", lies and so on in the present political context (not the least because discussion of that is against the rules of this sub-reddit, rightfully so), it's important not to overestimate the degree to which 20th Century publics routinely and extensively fact-check or verify all ideas presented to them. That would be true even of more rarified publics--say, practicing scientists within academic institutions. There's been considerable unease in the last decade within the sciences as it is has become clear that many established or accepted research findings are actually difficult to replicate or seem to rest on sophisticated forms of statistical manipulation. So as we look back, we should be aware that we are evaluating less of whether a given society in any time or place actually routinely checks facts or verifies truths and more whether this is something that people believe is happening or should be happening, and what kinds of practices and institutions they believe should be assigned to that work--as well as what they understand "facts" and "truths" to be.

Broadly speaking, many intellectual and cultural historians, as well as historians of science, would argue that the 17th and 18th Centuries in Western Europe saw the creation of new ideas and institutions devoted to establishing standards for truths, facts, and measures, and for official verification of events and evidence.

This is not to say that medieval European societies had no such records or practices. The office of a notary--a government or church official charged with recording particular economic transactions, ownership of property and goods, payment of taxation, and so on--goes back to classical Rome and survived in some form through most of the medieval period in Western Europe. Parishes throughout Western Europe kept detailed records of some aspects of their parishioners' lives: birth, death, marriage. Wills and inheritances recorded the wishes of individuals as stated concretely on particular dates. Treaties and contracts stated agreed upon facts pertaining to the resolution of issues or obligations within (though of course some of these were signed under duress, and effectively sanctioned fictional or distorted accounts of various events, as they still do.)

Nevertheless, a lot of historical scholarship builds on (and often complicates) the proposition that in the 18th Century in Western Europe (and the colonial Americas), common understandings of the concept of a fact and the meanings and uses of truth began to shift substantially and that new institutions and practices devoted to the creation of facts and the formal evaluation of truth began to appear. Often this shift has been linked to or attributed to "the Enlightenment", but that's a disputed concept in its own right (Steven Pinker's dubious history notwithstanding).

Just as it is important to not overstate contemporary devotion to a single or simple idea about facts or truth, you could easily overstate "Enlightenment" thinking about conforming to contemporary views of rationality or truth. Isaac Newton famously was as (or more) interested in alchemy as he was in calculus or gravity. At least one historian has argued that to understand Galileo as driven by the pure pursuit of scientific truth (and punished by the Church for it) is to misunderstand the nature of Galileo's motivations and outlook--Mario Biagioli observes instead that Galileo was primarily trying to produce work that was aesthetically and intellectually pleasing to his patrons (much like Renaissance artists did) and was primarily operating within the context of courtly politics in Renaissance Italy. The Royal Society of London, often seen as a key force in the birthplace of scientific thought, undertook all sorts of investigations (and claims) that seem at best eccentric from a contemporary standpoint. Enlightenment thinkers were frequently completely satisfied with the truthfulness of claims and observations that we today would regard as obviously and entirely fantastic and imaginary, particularly when they came from European travellers to non-European parts of the world. In many common fields of thought and investigation, fabulistic or imaginary claims remained normal and acceptable in public life until the late 19th Century--history included.

Nevertheless, historians like Mary Poovey, Steve Shapin and others have argued that ideas resembling the modern conception of "facts" and "truths" (scientific and otherwise) appeared in new forms and guises during the period that the OP is asking about. At least some of those ideas included procedures for testing and evaluating the accuracy of interpretations of truth and controlling or adjudicating what was judged to be factual. This was also an era of judicial and political reform that often had similar aims--to more thoroughly evaluate the status of evidence before a court, and to more thoroughly regulate what a government considered to be true and how it collected information throughout its territory. In the late 18th and 19th Centuries, both governments and private institutions built up archives and libraries and looked to systematize the information they contained; the Enlightenment thinker Diderot tried to create a comprehensive Encyclopedie that would contain all valid knowledge and systematize it. Scientific societies tried to create regulated schemes for naming and describing the world and rules that would govern future practice.

The work of the philosopher-historian Michel Foucault and historians strongly influenced by him takes these claims one step further, but that's a bigger and more complicated discussion.

Mary Poovey, The Invention of the Modern Fact, 1998

Steve Shapin, A Social History of Truth, 1995

Lorraine Daston, Histories of Scientific Observation, 2011

Pamela Smith, Ways of Making and Knowing, 2017

Brian Ogilvie, The Science of Describing, 2008

Jacob Soll, The Information Master, 2007

Marcus Krajewski, Paper Machines: About Cards and Catalogs

Peter Burke, Social History of Knowledge: From Gutenberg to Diderot

Ann Blair, Too Much to Know: Managing Scholarly Information Before the Modern Age