Are dimmis allowed to propagate their faith?
Are dimmis allowed to debate with Muslims regarding religion?
According to "reliance of the traveler" (translation by Nuh Ha Mim Keller), the matter seems a bit involved. The ahl al-dhimma, i.e. the dhimmis under the protection of the islamic state (this contract is the dhimma), that have entered into the dhimma face certain restrictions, whether they are expressly stipulated in the dhimma or not. In o11.5, the "reliance" says the dhimmis (changes and ommissions in  are mine)
(6) are forbidden to [...] (A: ring church bells or display crosses,) recite the Torah or Evangel aloud, or make public displays of their funerals and feastdays;
(7) and are forbidden to build new churches.
Those acts would violate the dhimma, whether it says so in the dhimma expressly or not. Now while the dhimma does not automatically disallow prosyletization per se according to this, the "reliance" says in o11.10
The agreement is also violated [...] if the state has stipulated that any of the following things break it, [...] though if the state has not stipulated that these break the agreement, then they do not; namely, if one of the subject people: [...]
(3) leads a Muslim away from Islam; [...]
(5) or mentions something impermissible about Allah, the Prophet (Allah bless him and give him peace), or Islam.
The consequence of the violation of the agreement is discussed in o11.11:
When a subject's agreement with the state has been violated, the caliph chooses between the four alternatives mentioned above in connection with prisoners of war (o9.14).
Those alternatives are (see o9.14):
[...] death, slavery, release without paying anything, or ransoming himself in exchange for a Muslim captive held by the enemy.
If the prisoner [my comment: o9.14 talks about prisoners of war, so in analogy this part may or may not apply to a dhimmi who violated the dhimma] becomes a Muslim (O: before the caliph chooses any of the four alternatives) then he may not be killed, and one of the other three alternatives is chosen.
In general, the issue does not seem to be amenable to a blanket answer by default, certainly not one covered by ijma; the book Religious Pluralism and Islamic Law (in particular pp. 124 ff.) gives an overview over the relevant positions and relates of the shafi'i position what I outlined above. As a matter of historical fact however, whether it was by law due to the legal opinions dominant in a madhhab or by the stipulations of the dhimma, dhimmis have never been allowed to prosyletize among muslims in the history of islam. As for debates, I suppose that would be considered spreading their views about - how would they argue if not for their position? - and would thus also have been outlawed.