The question for trespassing is whether the trespasser has been adequately notified of the possession interest that another person has in the property. Although it is legal to evangelize door-to-door without a permit, most jurisdictions will require you to leave promptly if the owner asks you to leave. In the absence of such verbal requests, you have a legally protected interest in engaging in conversation with your neighbors.
Based on the law as I understand it, it is likely that your jurisdiction will recognize a right to proselytize to your neighbors door-to-door regardless of a no trespassing or private property sign. However, for the sake of certainty, here’s a little bit about trespass law. Typically, states have statutes that define what it takes to make a sign “conspicuous” enough to create grounds for prosecution. Such laws include specifications for the thickness of markings and the sign’s distance from the ground. Arkansas statutes have established that a person who enters or remains unlawfully on property has committed trespass, which means that the person must enter or remain in or upon the premises when not licensed or privileged to enter or remain in or upon the premises.
Communication is key. Arkansas law provides that if the property is unimproved and apparently unused, and is not fenced or otherwise “enclosed in a manner designed to exclude an intruder,” the person entering the property does so with license and privilege. Generally, however, if a person unlawfully enters property that has been improved, he has trespassed regardless of a sign on the premises devoid of some religious purpose for his visit.
Thus, there really is not a significant difference between the “No Trespassing” sign and the “Private Property” sign. Both would communicate to any potential intruder that the property he is about to enter is presently being possessed by another person and thus would likely be sufficient to provide grounds for claiming notice in court if a property owner should choose to bring a suit.