First of all, the statutes consider this a petty misdemeanor. The wording is as follows:
NMSA 1978 Section 30-7-4
E. Carrying prohibited on private property.
In addition to other limitations stated in the act, a licensee may not carry a concealed handgun on or about his person on private property that has signs posted prohibiting the carrying of concealed weapons or when verbally told so by a person lawfully in possession of the property.
As far as the owner having to ask you to leave, that is up to the owner. They can call the Police and have you arrested. They can just ask you to leave. Or, they may just ignore it. Hopefully nobody would know you are carrying concealed in their private business. I do carry concealed sometimes (never in an alcohol establishment- that is a felony) in malls and theatres where it is posted, but the chances of someone knowing I am carrying are very slim. However, if you are asked by the owner to abide, you had better do it unless you want to be arrested. The law is on their side. You can ignore the sign, but once confronted, I hope they just ask you to leave and you should.