A Flight Instructor reported two NMAC's while on downwind in the Tower VFR pattern.

Narrative

Coming back in from the mountain pass; VFR in bound full stop at ZZZ. Tower told us to enter 45 to the downwind for right traffic runway XXR and to follow traffic in front of us (Cherokee). As we were starting our turn to the downwind the Archer that was following us in was told by Tower to follow the traffic that we were told to follow (Cherokee). They (the Archer) then cut us off getting so close that I could see the people in the aircraft. At this point I took flight controls and initiated a 360 degree turn in the downwind for spacing. Then a different controller came on to apologize for the miscommunication and let us know they (the previous controller) were training. Later in the downwind Tower again initiated a 360 for spacing to which the Archer did a turn 540 degrees. When exiting the turn; they looked head on to us; resulting in us having to again take evasive action. Tower then told them to fly base along the freeway which then allowed them to change course. Throughout this interaction there were no traffic alerts issued by the tower. Final to landing no incidents occurred. I also called the tower manager following this occurrence. They again apologized and suggested we speak up when we believe instructions are incorrect and also for all to maintain more vigilance in this crowded airspace