The restaurant is under the supervision of Rabbi Zushe Blech, a Monsey rabbi who is widely known in kashrus circles. I guess they describe it as a “private hashgacha” to distinguish it from a national hashgacha like the OU or a regional one like RCBC.
Suffern is basically Monsey, just like Airmont, Spring Valley, Wesley Hills, etc. Different local government or ZIP code but it’s all the same community.