Great question.
The first thing you do is actually buy the suppressor. You pay up front for it, decide how to own it. Decide wisely, you own it. Costs 200 to transfer it to someone else in addition to the paperwork.
How to own it is the part where there are options. It is transferred to you from the dealer on a Form 4, there is a 200 dollar tax each time a Form 4 is submitted. So if you buy 2 suppressors, its 2 tax stamps. Finger prints and photos are submitted with the form 4.
Personal- You put it in your name. You submit your prints and pic. You have to be present when it is being used. When you die, your heir can do a tax free transfer on a Form 5. Can be confusing to the people that get your stuff.
Corporation- Form a corporation. It can own them. I dont see it often and the couple I have seen were people paying another 200 per item to get them out of their corp because they were closing it or selling it with a corp asset. (ie beach house in Rocky Point)
Trust- My preferred method, but I dont push it on anyone, just remind them of the conversation....I told you so.... when you buy on a trust. Each trust is written differently but the one I use allows me to have multiple users and actually assign temp members to the trust. But lets go with the Trustee/Responsible person/persons. They have to do fingerprints and submit a passport pic when they buy an item. The tax stamp is paid and the paperwork is submitted.
In all cases, a copy is required to be sent to your county LEO. They dont have to approve it, but they get notified.
In the olden days, before computers, the only way to send in forms was doing paper forms. You still can do them that way. It takes longer and requires mailing printed forms and actual ink prints and photos. Most people do not do them this way currently, unless buying a NFA item from an individual. (ie Machine gun, destructive device or rare suppressor)
E-file The modern way. Your dealer files the form 4 for you electronically, you docusign them. You set up an account with the ATF. This account gives you a personal login and private personal identification number. Once your forms are submitted, the ATF send a notification to have you certify that yes you are buying this item. Once that is done, they process the forms, and once they are approved you and your dealer get an email with a picture of your approved stamp on it. At that point you can go pick up your item. You do a 4473 to have it transferred but they do not normally have to do another background check.
That is the basics at this time. Things change.....