From the preface of book “Praying with fire” (a great book btw)
The mabit writes that tefillah is a mitzvah in the Torah, “To serve Him with all your heart”…. The gemarah says “it means tefillah”…. The form of the mitzvah of tefillah is to express Hashem’s praise and to ask for our various needs, because by doing so, we affirm that the fulfillment of our needs lies in His hands alone….
The Chovos HaLevavos writes that the purpose of tefillah is not to change Hashem’s mind but rather to bring us to the realization that our fate is completely dependent on His will, that we can only survive through His mercy. The very act of prayer, however, elevates, exalts, and transforms us, so that we are no longer the same people we were before……
Rav Eliyahu Lopian said “When people pray for another person, they become like his talmidim in that he is causing them to gain merit.”
In other words, whenever a person causes a good thing, whether consciously or unconciously, he gains merit for causing it.
When we pray for a sick person, we draw closer to Hashem, and we become transformed through prayer. The sick person caused this transformation, and the merit for that goes for him. The more people that pray for him and are elevated, the more merit he accumulates.
So in answer to your question, do you yourself benefit from saying tehillim for another person? Absolutely!!!