When customers use PayPal; Square Cash; Facebook "Messenger" or other peer-to-peer payment methods - Who is responsible to handle disputes arising from these payment methods?

Is it the customer's bank or the Service Provider?

Depends on what the dispute might be.

As rlcarey stated, there can be a lot of variables to consider and the devil is in the details. It may be prudent to test drive one of these services yourself and print/review the terms of service; some of these TOS agreements I've seen in the past will include directives to report issues to the service provider. However, expect the service provider to roll over and tell your customer or cardholder to report any issues involving unauthorized activity to the FI if they know it takes the liability off of their shoulders.

We are considering Square - do you know if there are any banking regulations we need to follow?