The government or the official institution concerned with Trevi Fountain to collect the coins give them Caritas charity to benefit the community and therefore to give the wish a value, plus the estimated 3,000 Euros are thrown into the fountain each day, and this amount help minimum 3 poor family daily.
Many visitors to Rome and the Trevi fountain are Catholics and I believe they would prefer the money to go to a charity that has the same ideals. The Caritas charity is international, so donations made through the fountain can help affect change in the counties of all who donate.
The Caritas charity is responsible for creating food and social programs all over the world, and with roughly €3000 collected from the Trevi Fountain every day, Caritas is able to help thousands of people in need. I think that it's great that a popular tourist tradition can transition from a mere novelty to an actual way to insight change in the world.
I do believe that the coins that are tossed in Trevi Fountain should go to the Caritas charity. It is estimated that over 1.26 million dollars are tossed into the fountain every year. This is a significant amount of money that can be used to fund a variety of charities including feeding and caring for children.