Our DD did this at four months as well and I was concerned but DH was not.
What I learned is that in some countries babies are still put on their stomachs to sleep from the get go. I have a friend who was raised in a different country, who lives her now and she had her baby on her belly when she was born. That said, I do believe that at four months, the SIDS risk is lower.
We let our DD sleep on her belly when she turned herself over. We did have her on a wedge under the sheet of the mattress until she was 8 months.
I like th eidea of the safety monitor, if it makes you feel more at ease. We did not have one.