At first glance there are 2 possibilities, שמיעה or ,תקיעה. The mitzva is either to hear the shofar or blow it.
This seems to be a machlokes harishonim. The Rambam and others emphasize that the mitzva is שמיעה and therefore the beracha is לשמוע קול שופר. The Rosh on the other hand quotes Rabbenu Tam that the beracha is על תקיעת שופר strongly implying that the mitzva is the תקיעה.
The acharonim point out that both opinions are quite difficult.
The acharonim ask a number of questions on each opinion. If the mitzva is שמיעה then why does the תוקע have to be מכון to be מוציא you? After all, you heard the shofar, who cares what the blower's kavana was. On the other hand if the mitzva is the תקיעה then how can somene else blow for you? We know that by a מצוה שבגופו you can't appoint a שליח. For example, someone else cannot put on tefillin for you or sit in the succah for you or shake a lulav for you. If so, how can they blow shofar for you? Also, according to R' Tam why does the mishna say that if you blow in a בר and you hear an echo you are not יוצא, why not? After all you blew the shofar.
Based on the above and more the acharonim say that shofar is a hybrid, both the תקיעה and the שמיעה are part of the mitzva. Therefore, you need both aspects to be יוצא.
We are left with 1 question how can someone blow for you it is a מצוה שבגופו?
The answer may be as follows. Sofar is considered by the gemara to be avodas pnim and is considered to be tefila. The Gemara says that we say מלכויות to crown hashem, זכרונות so that he remembers ובמה בשופר. The shofar is a vehicle of tefilla and therefore maybe just like I can be מוציא you in tefilla through שומע כעונה, so too I can be מוציא you in תקיעת שופר because it is tefilla.
The machlokes about the beracha would then seem to be which of the 2 parts of the mitzva is the עיקר , more important. Although both are needed, the beracha needs to focus on one and the machlokes is which one should we pick.