"To him who is able to do far more than we can ask or imagine, according to his power that is at work within us, to him be glory in the church and in Christ Jesus throughout all generations, for ever and ever! Amen. "(Eph. 3:20)
Consider that when you ask the priest to celebrate a Holy Mass for a specific intention, to which it also adds an offer (which, mind you, is not a payment because nothing can be sufficient to repay the value of a Mass, but it is just a handout to join the Holy Mass), we're actually asking that all the merits of the Passion and Death of Christ, then all the blood spilled, all the tribulations suffered internal and external, everything, absolutely everything is applied to that specific intention. And as if Jesus was really dead only for that specific grace that we are asking only for that person that we put into the heart. Do you think that there could be an act of charity bigger than this? or a prayer bigger than this? St. Philip was quite right to say that "in prayer we ask God the graces in the Mass we force him to give them to us"
When you ask the priest to celebrate Mass for the needs of a living person, whatever it is, the important thing is to point out that the person in question is alive and if you do not want to explain (for many reasons) the details of the request is enough celebrate it, "according to the intentions of those who require it", then the applicant will be to put his intention into the Heart of Mary because she gives it to Jesus during Holy Mass.