There are a number of factors at work here. They depend on the type of peasant, what industry they laboured in (or if agricultural what crop), what economic status they might have had, and what period they lived in, and finally where they lived.
Peasants were rarely rooted to the earth they tended (admittedly those in remote regions might have far less exposure than those near urban or market centres). In certain periods (such as the Black Death) peasant labourers could demand high wages to work land and could survive working seasonally (P.J.P. Goldberg has called this the 'Golden Era' of peasant wages - especially for women). Peasants might, especially towards the late end of the medieval into the early modern period, travel to urban centres to apprentice or simply work as household servants. If they lived near major land routes or strategic locations then peasants might be exposed to more news through travellers and word-of-mouth. They might travel themselves on pilgrimage. If they were Christian then it was likely that they would attend communion frequently (if not every week) and therefore could look to their parish priest for any new information. At the sharp end of things politics, wars, and the monarchy might come to them - armed and rapacious in action if not intent. Or on a less violent note (at least locally) an Archbishop might appear recruiting for the crusade (see the Iterarium Cambriae (Journey through Wales) of Gerald of Wales for one such example) - or simply on a routine inspection of his diocese. Moreover, in the early through central middle ages many king's courts were itinerant (ie. they travelled) in order to consume the taxes they levied (food render). Or a special occasion might call all the peasants to a regional centre (as occurred in 1209 when John I of England summoned everyone to do homage to him and his newly born son Henry [III]). This in particular was an extraordinary event but would have been excellent for spreading news and gossip. [see J.R. Maddicott's excellent article on this event: 'The Oath of Marlborough, 1209: Fear, Government and Popular Allegiance in the Reign of King John', English Historical Review, v.226, n.519 (Apr., 2011), 281-318.]
These factors all would have affected how exposed a peasant might have been to the 'outside' world. A final word on period: peasant experiences, literacy, communication infrastructure, and the spread of Christianity would all affect these factors and the period and geographical scope of your question is very broad (1000+ years across a massive geographical location). The experiences of a peasant in the Shetlands in the year 500 CE would be very different to those in Lorraine in 1400 CE. I am not trying to harangue you but simply point out that you will get a better and more detailed answer if you can specify when, where, and (optionally) why you are interested in this question.
I had a scan through the backlog and no one has answered this question in any type of depth or satisfactory academic detail so I do hope that someone does come along with more comprehensive knowledge of the period than I! That said no one will probably have the expertise to cover the entire period.
In lieu of any semblance of holistic discussion on my part, here is a random historical anecdote which might be of interest:
In the early fifteenth-century peasants were not only aware of goings on around them but took actively partisan sides:
Such rivalries certainly existed towards the end of the Middle Ages and the 'identity' needn't even be national. Here is an example from the fifteenth-century:
Joan of Arc is undergoing her third public examination (Saturday 24 Feb. 1431) at the Rouen Trial (February-May, 1431, after which she was condemned as a schismatic). Here is a translation (source below):
Asked if the people of Domremy supported the Burgundians or the other party, she replied that she only knew one Burgundian there, whose head she would have wished to be cut off, that is, if this had pleased God.
Asked if [those of] the village of Maxey were Burgundians or enemies of the Burgundians, she answered they were Burgundians.
Asked if the voice told her in her youth to hate the Burgundians, she answered that since she knew that the voices were for the King of France, she did not like the Burgundians. Item, she said that the Burgundians would have war if they did not do as they should; and she knew it from her voice.
Asked if she received a revelation from the voice in her youth that the English should come to France, she replied that the English were already in France when the voices began to come to her.
Asked if she was ever with the little children who fought for the party that she supported, she answered not as far as she remembered, but she certainly saw some [of them] from the village of Domremy who had fought against those of Maxey, returning sometimes wounded and bleeding.
Asked if in her youth she herself had any great desire to pursue the Burgundians, she answered that she had a great wish and desire for her King to have his kingdom.
Now the issue was clearly a hot-potato in the Lorraine village of Domremy. Depending on exactly when this had occurred the villagers may have known about the murder of the Duke Charles of Anjou (1410) at the hands of the Duke of Burgundy, or they might have heard rumours about the murder of the Duke of Burgundy (1417) at the hands of the Dauphin Charles (allegedly). What they would definitely have known is that the Duke of Lorraine was on the side of the Dauphin and the Duke of Burgundy was in alliance with the English. This fight may have been born out of regional frustration or it might have been inherently local. The questioning may have forced Joan to frame her answer in these factional terms or it may have been engendered by her own personal hatred of the Burgundians (one of whom had captured her and sold her to the English - Jean of Luxembourg).
If I get a chance later I'll try and add a comment about my favourite medieval tradition the 'charity-ale' which was a demonstration of reciprocity and obligation at a very localised level - but also leads into the topic of pubs as a centre not just for enjoying news and fine craft ales (because they were largely all craft ales) but brewing rebellion!
A list of some interesting sources:
Judith Bennett, ‘Conviviality and Charity in Medieval and Early Modern England’, Past & Present, No. 134 (Feb, 1992), 19-41.
Christopher Dyer, ‘The Social and Economic Background to the Rural Revolt of 1381’, in The English Rising of 1381, Cambridge, 1984, 9-42.
P.J.P. Goldberg, Medieval England: A social history, London, 2004.
Mavis Mate, Women in Medieval English Society, Cambridge, 1999.
Craig Muldrew, The Economy of Obligation: The culture of credit and social relations in early modern England, London, 1998.