The Hachamim derived from the Pasukim that one may only fulfil the Misva of Sisit with a Tallit that belongs to him. There are a number of scenarios in which this rule causes a Halachic dilemma. If someone asks his friend to borrow his Tallit in order to pray and fulfil the Misva of Sisit, he cannot fulfill the Misva if his friend merely lends him the Tallit. Interestingly, the Mishna Berura (Rav Yisrael Meir Kagan, 1839-1933) rules that in such a case, it is assumed that the friend gives him the tallit as a gift on condition of return, in order enable him to perform the Misva, as per his request. Therefore, he may recite a Beracha on the Tallit.
However, there are cases in which a person asks to borrow a Tallit for purposes other than fulfilling the actual Misva of Sisit. For example, a Kohen needs a Tallit to cover his hands for Birkat Kohanim, a person who needs a Tallit to be called to the Torah for an Aliya or a person who already prayed but wants to sit in Shul with a Tallit like everyone else. In all of these cases it cannot be assumed that the friend is giving it to him as a gift, since his intent is not to fulfil the Misva. Therefore, because of the doubt, one should not say a Beracha in such cases.
All of this applies to borrowing a Tallit from private individuals. However, when a person takes a Tallit belonging to the Bet Knesset, which is communally owned, it can always be assumed that the Tallit was donated with the intent that all who wear it have temporary ownership so that they fulfil the Misva. Therefore, one should always recite a Beracha when using the Shul's Tallit.
The Shulhan Aruch rules that one may even borrow a Tallit without the owner's permission. If someone needs a Tallit, he may go over to someone else's Tallit bag and use their Tallit, because there is an inherent assumption that the owner wants Misvot to be performed with his possessions. The Rama extends this principle to borrowing Tefilin, as well. However, he restricts borrowing books without permission, as the owner may be concerned that the pagers will rip.
The Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) in Parashat Lech L'cha, as well as Rav Yishak Palachi in his "Yaffe La'Lev," rules that this principle does not apply today. Nowadays, most people are particular about letting others use their Tallit without their permission. They may be concerned about someone else's perspiration, saliva or germs coming in contact with their Tallit. It can be assumed that people are even more finicky about letting others use their Tefilin without permission, because they come in direct contact with the body. The Ben Ish Hai adds that he himself is included in this majority who would prefer that people not use their Tallit and Tefilin without permission.
Moreover, it is possible that even if a person asks for permission to use a Tallit that the owner is not agreeing with a full heart, but was embarrassed to refuse. If so, one could not recite a Beracha on that Tallit. It is for this reason that the Halacha Berura says that Hacham Ovadia was careful never to ask anyone to borrow their Tallit. If Hacham Ovadia was concerned that the Tallit may not be given wholeheartedly, how much more so we should be concerned. It follows that the Shamash has no right to offer other people's Tallits to newcomers who lack one. Rather, they should give them a Tallit form the communal bin.
If a person asked to borrow a Tallit to perform the Misva, he recites a Beracha. If he asked for the Tallit for some other reason, he should not say a Beracha, unless he borrowed a communal Tallit owned by the Shul.
One should refrain from borrowing someone else's Tallit or Tefilin without their wholehearted consent.