We don't offer a recording feature but we find there a lot of free solutions our providers use.
If you use a Mac you can use quicktime to record. Screencastify is a free chrome plugin.
You can record a session with Camtasia that works in parallel with doxy.me. Then you can save it to your computer or local drive. Here's the link to a free trial of Camtasia.
It is a feature we are planning to implement eventually.