It seems possible to send peer to peer payments using Adaptive Payments. For example, between two email addresses. This is free using the paypal app but is it free to do with the API?
Exactly, The PayPal Adaptive Payments APIs allow you to set the payment type as PERSONAL so that if the user's payment is funded by a bank account or by PayPal balance, there wouldn't be any transaction fee.
Check the API specs here: Adaptive Payments - PAY