In order to answer that question, we first need to decide if Beowulf, in its present form, even existed in the year 1000. This used to be accepted; JRR Tolkien said that the poet probably lived in the first generation after the arrival of Christianity, so contemporary to the Venerable Bede, and he is looking back at the pagan heroic ethos of the poem. Since then, scholarly opinion has swayed back and forth; for a while in the 1980s and 90s it was popular to assign it a late date, in the late 10th or early 11th century, an overview of this can be seen in Frank 2007. However, more recent arguments again push an early composition date, prior to the 8th century (Neindorf and Pascual 2014; Neindorf, ed. 2014.).

Regardless of the date of composition, some version of the story must have been circulating throughout early medieval England. The story reflects very old and widespread oral traditions of Germanic references. Beowulf either references the same events as or mirrors episodes within numerous Old Norse texts, including Hrólfs saga kraka, Þiðreks saga af Bern, Grettis saga, Völsunga saga, and Skjöldunga saga. This is, on its face, a highly intriguing event; with such widespread parallels it has been tempting for a very long time to try and find some explicit parallel between Beowulf and any of these texts. However, many of these are based in conjecture and can't be verified (Fjalldal 1998). More productive is the assumption of a shared legendary milieu of which the texts are all products of; the texts then become analogues of each other, instead of sources (see Hobson 2019). This is helpful; it's clear that stories parallel to Beowulf, if not Beowulf itself, were circulating in Britain and the wider Germanic world.

Now, to finally address your question: Where would you hear Beowulf (or some legendary poem?) Well, most frequently, those would be the purview of a scop, or poet. These lays are attested indirectly, most frequently with some kind of zither or lyre (Ghosh 2016). Ghosh also outlines evidence that they were popular among clerics, and there is evidence that Charlemagne heard and learned this type of story; therefore, it's safe to assume it was popular among the upper class, to some people's regret (e.g. Alcuin, who hated that idea). Alcuin's letter specifically says "Verba dei legantur in sacerdotali convivio. Ibi decet lectorem audiri, non citharistam; sermones patrum, non carmina gentilium." [the words of god should be read at the gathering of priests. There, it ought to be heard from the lector, not the zither; sermons of the [Church] fathers, not songs of the Gentiles] (taken from Ghosh 2016, translation mine). This may indicate that there was reference to them in wider Mass, but that is a stretch. While scops were mostly likely attached to relatively affluent patrons, like a skald in the Norse context, they likely also travelled and could be seen in local peasant gatherings. It was at one of those, or at a seasonal festival (e.g. May day festivals), that a peasant around 1000 CE may have heard Beowulf.

Fun side note: Benjamin Bagby does performances of Beowulf with a reconstructed harp from that period. It's pretty cool.

The second question is easier to answer: If you wanted to read Beowulf.. we don't know where you'd go. There is only one manuscript of Beowulf, and if there were any other copies of the poem made, they disappeared without a trace. The extant poem is written in two different scribal hands, so there may have been an earlier exemplar that they were working from, but it's lost. The manuscript was composed somewhere around 1000 CE (give or take 30 years), and was split into two in the Middle Ages. According to the British Library's catalogue for it, the first half, the Southwick Codex, we know was at Southwick Abbey in the 13th century. The second half, the Nowell Codex, is completely absent until the 16th century, when Laurence Nowell owned it. Sadly, Beowulf is in the Nowell Codex, so its medieval provenance is completely unknown. We simply do not know where the manuscript was made, much less who it was made for, so we can't even narrow it down to a city or monastic scriptorium that had the manuscript at the moment of production.

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Sources:

British Library Manuscript Catalogue, Beowulf. accessed at: http://searcharchives.bl.uk/primo_library/libweb/action/dlDisplay.do?vid=IAMS_VU2&docId=IAMS040-001102971&fn=permalink

Fjalldal, Magnús, 1998. The Long Arm of Coincidence: The Frustrated Connection Between Beowulf and Grettis Saga. Toronto: University of Toronto Press.

Frank, Roberta, 2007. “A Scandal in Toronto: The Dating of Beowulf a Quarter-Century On." Speculum 82, 843-64.

Ghosh, Shami, 2016. Writing the barbarian past: Studies in early medieval historical narrative. Leiden: Brill.

Hobson, Jacob, 2019. "An Old Norse Courtly Analogue to Beowulf." Neophilologus 103, 577–590.

Neindorf, Leonard, ed. 2014. The Dating of Beowulf: A Reassessment. Anglo-Saxon Studies 24 Cambridge: D. S. Brewer.

Neindorf, Leonard and Rafael Pascual, 2014. "The Language of Beowulf and the Conditioning of Kaluza’s Law." Neophilologus 98, 657–673.

Tolkien, J.R.R, 1936. "Beowulf: The Monsters and the Critics." Proceedings of the British Academy 22, 245–95.