Prioritize user-added Flatpak remotes
If I have a remote added at a system level and a user level, I would expect the user-added remote to take precedence. This is how configuration files typically work, and more abstractly, if I've gone through the effort to add a remote, I'd expect it to be used by default instead of a system remote.
There's also the multi-user workstation case to think about, where by default, apps I install should not be installed for all users of the machine without more explicit consent.
What Linux distribution are you using, and what kind of package (RPM, deb, flatpak, etc.) are you experiencing the problem with?
This issue can be reproduced on Fedora Silverblue and Endless OS which both come with system-added Flathub. It may have overlap with #1754 (closed) and need design discussion.
Prior art
This is how elementary AppCenter behaves.
Android's Google Play Store only enables per-user app installation, though if the same app is installed by two users, their underlying files are shared.
I'd have to investigate how the macOS App Store and Windows Store behave on multi-user systems.