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.

NO, NO, NO. in the field training maybe, but in combat- are you crazy?. 1. Does the equipment meet ans is sanctioned by DOD. 2. If your personal equipment fails, both you and your NOK could be penalized.

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