Hi Geovid,
The PurePath Wireless bi-directional audio link is fully duplex, so yes, what you describe is possible.
There is no push-to-talk capabilities built in by default, but you might be able to do something similar in host controlled mode depending on what you want to achieve with this. With a fully duplex link the need for push-to-talk might be more to save power and not have the link up and running all the time?
Regards,
Kjetil