This is a very interesting question for Czech, where we use reflexive pronouns in case of co-reference and there is a clear distinction of co-reference and multiple reference in the third person
1) Já(i) jsem si vzal svůj(i) batoh - I(i) took my(i) bag
2) On(i) si vzal svůj(i) batoh - He(i) took his(i) bag
3) On(i) si vzal jeho(j) batoh - He(i) took his(j) bag
From a language user's point of view, I would say you can extend it frequently also to the first and second person plural.
4) My(já,Pavel) jsme byli na naší(tým) poradě - We(me, Paul) were on our(team) meeting.
5) My(já,manželka) jsme byli ve svém(já,manželka) domě - We(me,wife) were in our(me,wife) house.
I certainly cannot imagine using the reflexive svůj in the case 4).
However it is necessary to mention that the usage of the reflexive in the first/second person plural tends to get somewhat blurry in the last years and one can see the reflexive pronoun falling somewhat out of use even in case of a very clear co-reference (my personal suspicion is that this is an influence of advertising where they typically want to highlight you as the customer and your choices, which the reflexive pronoun does not really do).
Addendum: Also it surprises me that you would not permit *We(i) took our(j) bags. in English. I admit that without any pragmatic context, my preferred reading of We took our bags would be that with co-reference, however in a context of, say, a group of tourists, saying something like...
6) We (me,Paul) took our(our group) bags
...seem perfectly cromulent to me, but then again, I am not a native speaker of English.