We've moved a lot and as a result, have attended a number of schools. I disclosed once, but DS is very very mild. Even so, I had to submit a formal evaluation stating that he did not qualify for therapy, doctor and teacher evaluations from the last 3 years of daycare. Private not known for SN.
Second school I did not disclose, but like others said, it may let you into the school, but it shuts the door on executive functioning support. DS doesn't qualify for IEP or anything so I was ok at that point with the tradeoff.
One private school we toured, I asked the teachers re: ASD and they got this look on their face that made me instantly backtrack. We didn't apply there but I don't think they would have let us in after I said those words (not in DC).