Do most of Infantry units allow soldiers to use their own helmets and plate carriers?

In my experience this is typically not the case for helmets. You are welcome to purchase after market suspension systems for your helmet, but many units will have policies about using issued/authorized helmets.
Plate carriers tend to have more wiggle room within unit policies and SOPs. Often if a unit authorizes plate carriers they will authorize some off the shelf use, sometimes providing a list, as long as they accept issued plates. Other units may innsist that you use either army issue plate carriers or even IOTVs.
Of course YMMV and experiences are as varied as the units we serve in.

In general plate carriers are for a specific subset of any unit that has a specific need. Tank crew , IFV crew, the infantry teams climbing mountain tops in Afghanistan might all be authorized by unit leadership to wear a plate carrier and not full IOTV.
And all of those are going to be issued a plate carrier, that is what they will use. Now if leadership has authorized the use of a plate carrier and not full IOTV... Does anyone care if they are using the ACU pattern one issued, or a Multi cam version that holds the same issue plates in the same manner, covering the same area....nope not really.

Ive never see any SM, regular army, Guard, AF or Army reserve that was wearing a personal helmet over a issued one.
Has it happened? sure it has, right before they got body slammed for being stupid

No they do not. While most units allow non issued pouches and trucks, the body armor and helmet are vary rarely allowed to be purchased off the market.
There are several reasons for this-
Quality gear costs money, Joe should be spending $1400 just because he wants the wear the new badass ops core helmet. The one the Army gives us does its job just fine. And they are getting better all the time.
Knock off or improper gear is all over the place. Google Recondo vest and over half the results will be for airsoft or paintball use. There are too many companies selling gear for the Army to test and verify the quality and stopping power of body armor and the matching plates.
This quality control being up the last reason, investigations. If a casualty is taken, whether WIA or KIA, there will he an investigation. If the SM was not wearing Army approved helmet and body armor they will look to the commander for his risk mitigation plan. Most commanders do not want to accept this additional risk of non-Army approved plates and helmet.
Even units that once allowed non issued pouches can change depending on a new commander or cam. My last unit we could use whatever pouches we wanted that enabled us to get the job done. New BDE CSM and we were only allowed the Army issue garbage.
My advice is check with your unit, then omly purchase pouches that will help you do your job. At your level just start with an admin pouch for your status cards and a radio pouch for your radio.