Your presumptions are overstated, as we will explain, but your question is fascinating and prompts us to explore cases in which they do apply.
It is generally forbidden (as theft) to borrow other people’s objects without permission.[1] In a specific case in which we believe that the owner would want the borrower to take the object, the matter is more complicated.[2] Regarding an object that people generally are content for others to use, they may do so. Rishonim[3] write that people are happy to let others borrow an object that they will use for a mitzva.[4] The Shulchan Aruch,[5] following this assumption, allows one to borrow a tallit that he finds in shul. The Rama[6] expands this permission to tefillin, and elsewhere[7] he includes a lulav, as well. Indications are that the Shulchan Aruch and Sephardic poskim agree.[8]
However, the allowance to borrow religious articles without permission is not a blanket one. The Shulchan Aruch[9] makes a condition that one return the tallit folded if that is how he found it. The Magen Avraham[10] permits borrowing an article only occasionally and requires that it remain in the same building where it was found. The Rama[11] says that one may not borrow sefarim to learn from, as the concern that the sefarim might be ripped makes it unclear that the owner would be happy to lend them. The Pri Megadim[12] claims that the minhag is to allow borrowing siddurim in shul without permission, but he opposes the minhag based on the rulings of the Rama and others regarding sefarim.
The Aruch HaShulchan, a more recent leading posek who tends to be very minhag-oriented, turns everything upside down. Regarding a tallit, he not only interprets the classical leniency narrowly, but also observes that nowadays people are more particular about others borrowing their tallitot.[13] On the other hand, he says that people are no longer disturbed by others borrowing their siddurim and sefarim and that should therefore now be permitted.[14] Even sources from the period when sefarim were guarded more carefully assert that a talmid chacham who was given sefarim to watch can use them because of the assumption that an owner who did not explicitly state otherwise would allow a talmid chacham to learn from them.[15]
Thus, we find fluidity in the halacha based on the circumstances, and we do not find major differences between Ashkenazi and Sephardi poskim.[16] There is no uniformity in our day, and different places (especially yeshivot) have different practices. However, the consensus is that one may use another’s sefarim provided that he does not move them away from their place.
We will now re-ask your interesting question. Is it permitted for one who received a p’sak that it is permitted to borrow religious articles to make use of the property of one who received the opposite p’sak, and vice versa? Presumably, the reason there is a machloket on this issue is that owners’ intentions are difficult to gauge and/or due to differences regarding the general methodological question of how much to adapt classical rulings when recent observers sense that the situation has changed somewhat. In standard cases, these questions are for the borrower and his rabbi to determine and do not depend on assumptions regarding a specific owner/lender. On the other hand, if a certain ruling becomes widely accepted among a particular group, then the p’sak becomes a self-fulfilling prophecy, itself creating a minhag to allow or forbid others’ use. In other words, if all know that in a particular locality permission to use someone else’s sefarim is assumed, then even one who personally is inclined otherwise would probably decide to conform with the standard. Therefore, one who enters a yeshiva or a shul where there is a stated policy can assume that the sefarim owners who frequent those venues conform to that policy. Accordingly, the potential borrower should follow the owner’s presumed position, not his own. (Since an owner is always entitled to have others use his property according to his preferences, it is a good idea to look at a sefer’s inside cover, where some owner’s leave instructions as to whether and to what extent they allow others to use their sefer.)