If you're concerned about a web site silently having access to personal data from your phone, surely it doesn't matter whether that web site is storing it silently in your device's browser storage, or storing it on the web site's servers.

If, on the other hand, you're explicitly uploading or entering data on a web site, well then yes, the default assumption would certainly be that the data is getting saved on the web site's servers (if it's being saved at all). That's how the web has worked for a very long time, with the exception of cookies and a few other more recent but not widely known storage mechanisms which in most cases aren't considered sufficiently robust for persistent on-device storage.