Each of your profiles function as separate entities. If you swipe on a player using one of your profiles, it won't impact their status on the other profile. This means that if you decide, for example, to pass on a player as a couple using your Partnered profile, either of you could still like that player as an individual, using your individual Solo profiles.
Both players in a Partnered profile have equal control over it, so if you pass on someone with your Partnered profile, your partner will also see that person as passed on when using your shared Partnered profile.
If you only want to explore #open with your partner, you can choose to hide your Solo profile: navigate to the Profile tab on the Settings screen (tap the button in the upper left corner of the screen then tap the Profile tab), scroll down and tap the Solo/Partnered button. Here you’ll see a toggle beneath the image in the Solo Profile section, tap this to toggle your profile from Active to Hidden. (You can also choose to hide your Partnered profile in the same way—just tap the toggle in the Partnered Profile section!)
When a profile is Hidden, it can no longer be seen by anyone in the #open community when they browse through the app or search via hashtag or username. Setting a profile to Hidden will also disable the ability for that profile to find or view other new profiles in the app (we don’t want invisible lurkers haunting our community!) While hidden, you can still continue chatting with profiles you have already matched with, and the owners of those profiles can still see your profile.