good on ya for challenging your own perspective!
and let me point out the obvious: i do not speak for "anarchists", i speak only for myself. some other anarchists may agree with me, some may not. there is no singular "anarchist" answer to any question that i can think of.
the concept of "that is mine" (in some permanent sense of ownership or property) begins to fall apart when you look at material objects as nothing more than tools to be used in meeting your needs and desires. while i am using that object (or idea, for that matter), i might consider it "mine", but i would only think of that as an explicitly transient condition, for the duration of my using it. so the terms that seem relevant to me are ones like possession and usage.
in this world, a primary concern for me (and no doubt many others) is having a space where i can be alone and free of all others when i want. if i have created that space, or otherwise found it available to me - ie, not being used by others - then i would again consider it "mine", for as long as i am using it. a huge challenge there is, what if i am not there *all* the time, but choose to come back to it? if someone else comes along while i am gone and sees that it is not being used, they might well consider it available and take it for their own use. and i have no problem with that. what i would probably do is leave a note (or some clear communication) that i will be back - say, by winter - and that i expect the space will be used by anyone that needs/wants it, until such time as i return. then, when i return, if they refuse to leave (assuming i still want them to) we will have a conflict. between us alone, 2 (or however many) individuals faced with a conflict that must be resolved without relying on a state or other outside mediation.
that scenario kind of touches on the huge question of "hoarding". if i grow a shitload of potatoes, and i am saving them to sustain me for the winter, what about "sharing" with others that may not have any? i surely feel no obligation to do so. but that doesn't mean i wouldn't. it all depends on that situation. if someone comes to me and demands that i share them, or tries to "guilt" me into sharing them, they are not likely gonna get shit. or at least, there will be some definite conflict. if they approach me as another autonomous individual that is hungry and would appreciate it if i could spare a few potatoes, i am likely to be very generous. perhaps, even, they may have something (skills or knowledge, if not material objects) that i would find useful or desirable.
one area where i have major issues with ancaps in these kinds of discussions is their inability to view these kinds of situations outside of an economic perspective. every interaction must be funneled through the filter of economic transactions. every thing and activity must be assigned a value (even if the "producer" of that thing/activity is the sole assigner of the value). that is the dominant paradigm in the modern world, and it is one i seek to destroy at every opportunity.
i am definitely into mutual aid, but not based on value or enforced reciprocity.
