I cannot think of a reason a PDF reader should connect to a server.
Tandem Gander helps you riffle through your PDFs. It is designed to have no network connection, no account, no analytics and no ads.
It does keep a small cache of cover thumbnails for the shelf, and it does remember where you left off in each PDF you have opened. These files stay on your Mac. Losing the cache is a tiny inconvenience at most: the covers are remade from your PDFs, and nothing else in the app is affected.
Your reading positions, anchor pages and settings are kept in the app's own container on your Mac. Cover thumbnails are kept in the app's cache. The folder you choose for your shelf is remembered so it can be opened again next time. None of this is sent anywhere, and Settings › On this Mac shows every one of these places, with a button to reveal it in Finder.
If you turn on Settings › On this Mac › Keep a local diagnostics log, the app writes a plain-text log of what it did to its container on your Mac. It is off unless you turn it on, it is never sent anywhere, and the same Settings section shows you the file and clears it. Its size is capped; the oldest entries are dropped past the cap.
If macOS is set to share crash reports with app developers (System Settings › Privacy & Security › Analytics), Apple may forward a crash report to me. That is Apple's system, governed by Apple's privacy policy; the app itself sends nothing.
Tandem Gander is sold through the Mac App Store. Apple handles the purchase; I never see your payment details.