Usual disclaimer - What follows is my personal opinion. It is based on things I have studied in my medical training, but it does not constitute formal medical advice.
Whether or not you get these negative side effects depends, at least in part, on whether you are actually taking a progestERONE or a progestIN and they are not the same thing at all. That said no two people are the same either, so there is also the question of how well your body metabolises stuff... but leaving that aside for a moment:
Sadly because most trans people are not medically trained they end up thinking that they are on progesterone because the doctor has told them that, when actually what they are taking is a progesterone analogue (synthetic).
Progestins are the, so called, synthetics like MPA (Medroxy Progesterone Acetate Aka Provera) and Levonorgesterol They are used in natal females because of their ability to mimic the effects of progesterone on the uterus. These will be unlikely to benefit a trans patient and will also likely have some degree of androgenic effects as they are not real progesterone, and in fact are fairly easily metabolised by the liver into androgens. Thus if you are prone to Male Pattern Baldness they can indeed cause it.
However true micronised progesterone has been available for some years now and this does NOT cause the same problems. Its side effects are actually largely beneficial - so much so that I personally take it largely FOR the side effects!
As far as I can see the doctors that talk about negative side effects and lack of benefit of progesterones base their arguments on old studies done on the synthetics and thus, in my opinion, their conclusions are at best unproven and at worst completely erroneous.
I personally believe that it was true that taking synthetics offered at best very limited benefits to a transwoman, however taking micronised progesterone is a completely different story and I think does carry useful benefits. However at this point, as there are no formal randomised control trials comparing the two types (at least not that I am aware of) this is only personal anecdotal evidence and therefore not proof.