Public domain texts here means “out of copyright and free to use”. While ancients texts are themselves public domain, a modern transcription may not be, as it may amalgamate different sources, make textual choices, correct errors and otherwise apply a degree of editorialisation to the Latin. Thus to be safely public domain, a Latin text must be a literal copy of a printed (and / or scanned) source document that is itself out of copyright, that is published before 1920-odd in the USA. Elsewhere, the editor must have died typically 70 years ago.
Perseus is the main contributer to the Open Greek and Latin project. there are other repositories that it amalgamates though.
This makes pretty clear what their view of the copyright vs public domain status of any given text is. Nevertheless there are other contributors to that project, and it seems Perseus is available as CC-BY-SA (sometimes stated as cc-by-nc-sa, but the repository seems to say cc-by-sa).
As has been said this is almost certainly an overstatement of the real copyright position, where these are copies of public domain editions, as they often are - it is the initial editing that entitled these ancient books to a new copyright, not digitisation.
There are also texts at Latin Wikisource which are clearly understandable as Public Domain, as they are literal transcriptions of public domain books. However, many texts there are not always linked to their original transcriptions.
The Latin Library is a good source of material as mentioned and usually but not always have the source of its texts, and says that they believe the texts presented are public domain. It seems reasonable to assume that if the source is stated and is public domain, then the literal copy is also public domain.
As a third easy source, there are some Latin texts transcribed at Project Gutenburg.
There are also many Bible transcriptions some of which are clear about their copyright status. The Clementine Vulgate Project in particular marks their transcription as public domain; it has also been imported into Latin Wikisource. You can also find copies on GitHub and Bitbucket.
All this said, there is another question as to whether a computer digesting texts, training an algorithm and creating a result is a potentially infringing act. In my personal view, it (morally) should not be, so long as you have the rights to access the text in the first place, as it is analogous to reading and making notes from a book, neither of which are infringing acts, rather than anything like publication; but the fact that copyright exceptions exist in the UK and Europe to “allow” machine reading and processing in certain circumstances suggest that this is not settled outside of the US, where it does seem more likely to fall into fair use.