Unless you are a Public Match in DNA Relatives, your matches need to send a DNA Relatives Introduction in order to contact you. You will see these introductions in your 23andMe Inbox:
After receiving an introduction, you have the option to accept or decline contact.
If you choose to accept, you will be given the option of keeping your name and 23andMe profile private, or sharing your name and profile with the relative. If you accept contact, both individuals can continue communicating via message.
Some introductions will also include the option to share genomes (without health reports). You can accept contact without accepting the sharing invitation.
If you choose to decline the invitation, the sender will not be able to contact you through DNA Relatives. Declining an introduction is permanent, and the same match will not be able to extend you another invitation if you decline.