A guest can make an inquiry by using the "contact the host" link on your listing.
All you need to do is respond within the time limit. No obligation to pre-approve and no need to decline. BTW if it is a booking request, you must accept or decline within the time limit.
This is confusing. What "private messages"? And do you mean that people are contacting you through the Airbnb message system, requesting to book?
If so, that's a normal part of the process, and highly desirable because it gives you a chance to sense who's booking. And that's critical to prevent unsuitable guests from booking.
If it's by some other means, then how are they getting your contact information? Airbnb hides that from unbooked guests.
Double check and contact Airbnb before given out an information. Once guests start asking you for your personal phone number to conduct business outside Airbnb it’s a BIG red flag.