Offering more masses doesn’t give any more benefit to the one being prayed for - in some ways, offering masses is as much about us and our charity as it is the one being prayed for. As @Tis_Bearself has said, prayers are never wasted - but allocated by God as He sees fit. The late great Msgr Ronald Knox suggested that “extra” prayers, over and above what was needed, were allocated to those with no one to pray for them.
While an offering for the mass intention is of course welcome it’s definitely not compulsory - simony (the selling of holy things) is regarded as a very serious offence by the Church. So a priest is obliged to accept a mass intention and shouldn’t discriminate between paid intentions and unpaid ones.
I don’t see anything wrong (in principle at least) with offering a mass for multiple intentions - there is, after all, no limits to God’s grace; however there is a whole lot wrong with a priest keeping multiple stipends! My diocese requires its priests who receive a second offering on the same day to send it to the seminary in order to guard against the accumulation of (paid) intentions. My interpretation of this is, if I already have an intention for the mass I’m to celebrate (be it paid or unpaid), then any money I receive for the second intention goes to the seminary even though I’ll carry the intention over to the next mass. It’s hard to explain this so I usually just say something like “this mass is offered for X and we also remember in a special way Y”. This really only matters when the intention is date specific (e.g. an anniversary).
Finally, anyone can of course offer their own mass (the one they participate in) for a particular intention. I wouldn’t claim that this is any more or less effective or than a mass offered by a priest but, again, there’s the whole charity thing…