By default, you can use Zotero without logging in, which will cause it to keep everything locally. It's not obvious how you might go about partially syncing to the cloud, however.

In practice, though you shouldn't store the SQLite in a cloud sync service, setting up a symbolic link for the backing PDFs in e.g. Dropbox works quite well and gives you more control over your data syncing. Then, you only need to rely on their cloud for the paper titles and metadata.