We have really cute scrapbook-like cards with the babies first name on it. There isn't any other information on it; parents love it. I think the cards help parents understand that we see their baby as a person, not just a patient.
We also have the baby's id bracelet on the bed which is used by lab, etc. It's small enough that I can't imagine it would be much of a privacy violation.
It's funny, in a room full of babies, privacy violations occur all the time. We pull curtains, but they don't seem all that soundproof

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