How should we respond when the People of the Book greet us with salaam?

If the Muslim realizes that the kaafir has said to him al-saam
alaykum, which means may death be upon you, he should respond
in kind by saying wa alaykum (and also upon you).

If he realizes beyond a doubt that he has greeted him with
the greeting of Islam (al-salaamu alaykum), Ibn al-Qayyim (may Allaah have
mercy on him) said: if the listener realized that the Dhimmi [Jew or Christian]
has said salaam alaykum and he is sure of that, should
he say wa alayk al-salaam or just wa alayk?

According to the evidence and principles of shareeah,
he should say wa alayk al-salaam, because this is more fair,
and Allaah commands us to be just and to treat others well.

(Ahkaam Ahl al-Dhimmah, 1/425, 426).

Shaykh Muhammad al-Saalih ibn Uthaymeen (may Allaah
preserve him) said:

These people who have come to us from the
east and the west and are not Muslims, it is not permissible for us to initiate
the greeting of salaam with them, because the Prophet
(peace and blessings of Allaah be upon him) said: Do not initiate the
greeting of salaam with the Jews and Christians. (Narrated by
Muslim in his Saheeh).

If they greet us, then we can respond in a manner similar
to that in which they greet us, because Allaah says (interpretation of the
meaning):

When you are greeted with a greeting, greet in return
with what is better than it, or (at least) return it equally [al-Nisaa
4:86].

When they greet us with the greeting of Islam al-salaamu
alaykum either of the following two cases applies:

Either they pronounce the laam (l-sound)
clearly, and say al-salaam alaykum (peace be upon you),
so we may say wa alaykum al-salaam or wa alaykum

Or they do not pronounce the laam clearly,
and so they say al-saamu alaykum (death be upon you), so
we should say wa alaykum only. This is because the Jews
used to come to the Prophet (peace and blessings
of Allaah be upon him) and greet him by saying al-saam alaykum,
without pronouncing the laam. Al-saam means death;
i.e., they were praying against the Prophet (peace
and blessings of Allaah be upon him), praying that he would die. So the Prophet
(peace and blessings of Allaah be upon him) commanded
us to say to them wa alaykum.

Thus, if they say al-saam alaykum, we should
reply, wa alaykum, which means: and the same to you, may
death be upon you. This is what is indicated by the Sunnah.

But if we initiate the greeting of salaam with them, our
Prophet (peace and blessings of Allaah be upon
him) forbade us to do this.

(Majmoo Fataawaa Ibn Uthaymeen, 2/97,
98).

And Allaah knows best.

For more information, see Question # 6583.