Does the LWHEM or LDATA, require its own breaker?

Yes, both the LDATA and LWHEM require a dedicated breaker connection.  Stranded and non-stranded wiring should not be mixed under a Load Center termination screw.

The LDATA is wired to a dedicated non-smart 1-Pole Breaker and the Leviton Whole Home Energy Monitor (LWHEM) requires a dedicated non-smart 2-pole breaker.