There is no Mitzva to blow Shofar on Rosh haShana. Accordingly, there is no Bracha on doing so.
This is a Mitzva to hear the Shofar being blown, and there is a Bracha said before doing so ("...and commanded us to hear the blasts of the Shofar"). Everyone who is fulfilling the Mitzva should say a Bracha. In general, if multiple people are doing a Mitzva contemporaneously, one can say the blessing on behalf of all of them (OC 8:5) and this is commonly done in some communities for publicly performed Mitzvot such as reading the Megillah, hearing the Shofar, counting the Omer, and chanting Hallel.
Additionally, one who has fulfilled his obligation in a Mitzva can say the blessing on behalf of others who have yet to fulfill there Mitzva ("יצא מוציא"). The Rama mentions a custom that the one blowing the Shofar should utilize this option when blowing for others after already fulfilling his own obligation, but most Achronim think this should only be done if the people who have yet to fulfill their obligation cannot say the blessing themselves (for whatever reason). (OC 585:2 with Mishna Berura)