To add to Keiths posting.
This assumes that PilotAware and ADSB/Mode-S have been setup to use the same ICAO number, this is what PilotAware uses to determine it is from a common source.
For the Mode-S transmission, as Keith says - this will NOT generate a proximity warning as it is superceded by the PilotAware/ADS-B Positioned Data.
Having said the above, there is a slight potential gotcha, Mode-S transponders are configured to respond to all interrogations, so if interrogated by a Mode-C interrogation, it will elicit a Mode-C response.
If the receiving PilotAware is also setup to receive Mode-C, then there is the potential to report the Mode-C as a Proximity warning - this is because Mode-C transmissions do not contain the ICAO code, so cannot be associated with the ModeS/ADS-B/PilotAware.
The software in PilotAware has been written in such a way to try to mitigate the above dis-association between Mode-C & Mode-S/ES, we call this 'coalescing', and it works very well using a number of techniques to omit the Mode-C Proximity data, by determining that it has eminated from the same source as the Mode-S/ES transponder.
We have spent a lot of time refining these algorithms to help present multiple data sources in the most efficient and non conflicting manner.
Thx
Lee