Start by opening your address books page in Thunderbird. Right click on the address book you want to share and click "Sharing..."

Click the button in the upper left-hand corner of the window that opens.

Enter the email address of the contact you want to share your address book with. Click "Add..." when finished.

Double click the name of the contact you just entered.

Here you can edit the permissions this contact has when viewing your address book. Your settings will vary based on how many privileges you wish to give this contact. Press Update when you are finished.