It is permissible for Muslims to greet each other on the day of id with specific greetings that have been reported from the Sahabah (may Allah be pleased with them). Responding to a question in this regard, Ibn Taymiyyah said:

"As for people greeting each other after the id prayer by, 'Taqabbalal-Lahu minna wa minkum (may Allah accept from us and you),' or, 'May Allah bring the id back to you next year,' etc. - it has been reported that some of the Sahabah practiced it, and the scholars permitted it. However, Ahmad said, 'I do not initiate saying it to anyone; but if one says it to me, I answer him. This is because responding to a greeting is obligatory, but to initiate the greeting is not a required Sunnah, neither is it prohibited; whoever does it would have an example (from the Salaf), and whoever does not would have a predecessor as well.'"

Jubayr Bin Nufayr said:

"When the companions of Allah's Messenger (Peace be upon him) met on the day of Eid, they would say to each other, 'Taqabbalal-Lahu minna wa mink (May Allah accept from us and you.)"

And Muhammad Bin Ziyad said:

"I was with Abu Umamah al-Bahili (Peace be upon him) and some other companions of the Prophet (Peace be upon him). When they returned from the Eid, they said to each other, 'Taqabbalal-Lahu minna wa mink.' "