Using mendeley and science papers from sci-hub in Germany
In Mendeley you can turn off file syncing if you'd like, by folder or for all files. Right-click any item under "My Library" (folder name, "All Documents", whatever you like), and choose Edit Settings from the context menu. You will be presented with settings that includes Synchronization options, including an option for "synchronize attached files", which you can turn on or off as it pleases you.

In terms of checking copyright or digital rights management schemes, I'm not aware of any reference manager that cares about this - but then back in the days of Napster (or even the early days of YouTube) no one cared either, right up until they suddenly did. The main difference is that most reference managers, including Mendeley, have extremely limited file-sharing ability, as well as limited space requirements, so they tend to be more like a cloud file backup service than a file-sharing service. Also, historically, "digital rights owners" disagreed that even having a digital file was legal at all, even if it was just a personal backup copy, and this made the 1990's a bit weird in retrospect - but should not be an issue with academic papers, which are mostly digital now anyway. And when services are only backing up files only you are using, and often accepting payment to do so, there is far less incentive for anyone to care about the provenance of the materials.

If you are concerned about the availability of the files in-case a service suddenly decides to start erasing files they can't establish you were licensed to obtain (which seems unlikely and technically difficult to achieve in any useful way, but I can't predict the future - and passing files around between colleagues and students/teachers is so common that it would be really disruptive to attempt), you can instead just have a system like Mendeley organize the files into a folder that you then backup/save with a backup service/system. But if you are concerned about being targeted for having such material at all, I'm not sure that just turning off a feature like sync would actually do anything to protect you at all, as there is no guarantee a software could not still scan files and report back to the mother ship whenever it pleased anyway. So I'll have to leave it to you to decide how big of a risk you think that is, and whether it is worth attempting to mitigate.