In order for a male to be entered into the covenant of Abraham, it is not enough for him not to have a foreskin. Rather, a circumcision needs to be carried out by a G‑d-fearing Jew, with the express intention to perform the mitzvah of brit milah.
So what is done for someone who was born without a foreskin or was circumcised by a doctor? Assuming that the lack of foreskin meets the standards of halachah, the solution is a simple and painless procedure, known as hatafat dam brit, by which a drop of blood is symbolically drawn from the circumcision area.
Today, this is generally done with a sterile needle and is quick and with no before- or after-care. While it is not accompanied by the blessings and prayers that are said during a full-blown circumcision, it is customary to celebrate with a feast.
*Gulp*
What happens when an adult gets circumcised? Is it a public ceremony?
At our son's Bris, the mohel took the opportunity to attend to the hatafat dam requirements of two gentlemen who accompanied him. They went to a secluded room upstairs; the procedure was performed discreetly and unobtrusively. I don't know if this is a usual practice, but it is clear to me that there is, at least, the option to make it a private matter.
I suppose there might be an option to make it a public (well, open to friends and community) ceremony if the celebrant should so desire; I have not heard of any such, but, then again, my knowledge in this area is very limited.