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I got some interesting discussion in the comments when I asked this question at Mi Yodeya.
The comments reflect various thoughts on whether it would have been practiced the same in Jesus' time as compared to today, according to the Jewish law (halacha), and why it matters (or doesn't). There were also some incomplete answers in the comments, such as quotes of prayers or blessings that might be said and indications of who is qualified to perform the circumcision.
There was a link for an informative article about how circumcision is practiced today. I accepted the answer at Mi Yodeya. You can view it at this link: