I’m thinking you likely will want to each set up a Category report sheet (or possibly a Tags report sheet) of some kind in each main sheet, and set it to the shared expense category, and then from a third Google Sheet, grab the data from the two other sheets, and then have a tab that sums up both what Alice paid and what Bob paid to find the household total, and then see who paid more and who owes what to whom.
Something like this to take the first step inside each personal sheet to get the relevant transactions: