Should I pay for private domain registration?

This question would be at your preference dependent upon how private you want the information to be.   All domain registrars register domains through Internet Corporation for Assigned Names and Numbers (ICANN).  The contact and registrant information for the domain registration must be accurate and made public.  The use of false information is a violation of policies which could lead to the termination of the domain registration.  With basic domain registration, the contact information you provide (Name, Address, Email Address, and Phone Number) are made public in the WHOIS database.  

Private registration, depending on the registrar, generally has a minimal fee.  Private registration practices may vary slightly depending upon your registrar.  In general, registrars for private registration will apply generic contact information to the WHOIS database for name, address, and phone number.  They will supply you with a private email address, which they often list as the generic email address they supply to the WHOIS database.  You should have complete control of the email private email address and how email is filtered through it.  For complete details of how your registrar handles private registration (where applicable), we recommend contacting your registrar or searching your registrars website for more information.

To see what kind of details are supplied, any WHOIS lookup site or utility can be used.  Try it out!  For an example, go to  https://whois.icann.org/ and search for appliedi.net.