Bottom line it depends what you're putting it on for. From Dose of Halacha:
..The Mishna Berura (14:11) writes that when one borrows a friend’s Tallis to Daven for the Amud, one should make a Bracha. The Aruch Hashulchan (OC 91:2) and Kaf Hachaim (OC 147:4) write that one who receives an Aliya (or any other Kibbud) should wear a Tallis out of respect for the Tzibbur.
The Mishna Berura writes that if the Tallis belongs to the Shul, one should make a Bracha, as it has shared ownership. He brings different opinions, however, as to whether one should make a Bracha on a friend’s Tallis. In Biur Halacha he writes that it better to borrow a friend’s Tallis to avoid the Safek of having to make a Bracha, though brings an opinion that when one just wears it for a short time and doesn’t place the Tallis over one’s head, one doesn’t make a Bracha. The Kaf Hachaim (OC 14:14 quoting the Ben Ish Chai) also writes that it is best not to say a Bracha when only wearing it for an Aliyah, etc.
R’ Shlomo Zalman Auerbach (Halichos Shlomo, Tefilla 12:20 n29) held that one only wears a Tallis when one’s called up on Shabbos and Yom Tov mornings. Many Ashkenaz shuls have the custom that those who get called up during the week and Shabbos Mincha also wear a Tallis. Due to the Safek involved as to whether one should make a Bracha, the custom developed that one doesn’t remove one’s Tallis straight away, thus necessitating a Bracha.
In conclusion, one always makes a Bracha when wearing one’s own Tallis during the day. The Mishna Berura brings various opinions about one who wears the Shul’s Tallis. While he writes that ideally one should make a Bracha, most follow the other Poskim, and don’t. If one does make a Bracha, he should wrap the Tallis around his head and keep the Tallis on for a bit longer. The best option for one who is called up for an Aliyah (on Shabbos morning) is to borrow a friend’s Tallis and not make a Bracha.