Which online payment gateways allow sending money, not just receiving it?

Unless I am reading your questions totally wrong, what you are trying to do, can be done with any payment gateway. What you are essentially doing here is creating containers (accounts) for both the receiver and the sender, and the ability to pay each other.

Venmo (product), Braintree (company), Dwolla should be able to accommodate this without issues. I would also like you to look at BancBox.

PayDivvy is another company you might want to look at, not too sure if that will fit in your solution.