user@example resides at my mail server.... I have mentioned same email address at common websites like freshbooks etc.

Now whenever these other common websites send email to other email domains residing on my mailserver, my mailserver won't accept it giving a bounce, "User not logged in".

How can I fix it ?

Not enough information is given to really assist. What email server, postfix, exim, ect are you running? Just guessing, it appears you are configured to smarthost to some other smtp server only and its not accepting anonymous email, or it thinks the example.com is a external smtp and not delivering locally. You probably should skim over the linode library, it probably has the fix. http://library.linode.com/email