So yes if you were arriving in to T5 on a domestic flight and connecting on to any other T5 flight you would not have to go through security, it is just a simple bp check and then straight up in to the airside departures area. It would be the same if arriving in to T2 on a domestic flight and connecting on to any other T2 flight.

The problem comes when you are changing to another terminal. Unfortunately there is no separate airside bus for domestic arrivals, so when you get the airside buses you are mixing with all those who have arrived on international flights. So when you get to T3 in your case unfortunately you will have to go through security at the connections centre along with all those who have arrived on international flights.