It is permissible for Muslims to greet each other on the day of Eid with specific greetings that have been reported from the sahabah (radiAllahu anhumma) and from amongst the Salaf.
Responding to a question in this regard, Shaykh ul-Islam Ibn Taymiyyah (rahimahullah) said:
“As for people greeting each other after the eid prayer by, ‘Taqaballahu minna wa minkum‘ (may Allah accept it from you and us) – it is reported that some of the shahabah 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 it would have an example (from the salaf) and whoever does not would have a predecessor as well.” [Majmoo al-Fataawaa (24/253)]
Al-Haafidh Ibn Hajr Al-Asqalani (rahimahullah) said: “We have related in “Mahaamiliyaat” with a chain of narration that is hasan, from Jabyr ibn Nufayr who said: ‘When the companions of Allah’s Messenger (sulAllahu alaihi wa salaam) met on the day of ‘eid, they would say to each other, ‘Taqaballahu minna wa minkum (may Allah accept from you and us ).” [Al Mahamiliyyat; hasan isnad; see Fath ul Bari 2:446]
Ibn Qudaamah mentioned in al-Mughnee (2/259) that Muhammad ibn Ziyad said: I was with Abu Umamah al Bahili (rahimahullah) and some other companions of the Prophet (sulAllahu alaihi wa salaam). When they returned from the eid, they said to each other, ‘Taqabbalallahu minna wa minkum (May Allaah accept it from you and us).”