Note: I've been holding this post for a long time, because at least three of Toronto's synagogues went through a search for an Assistant Rabbi this past year, and I did not want to be misunderstood as commenting on any of those processes. This post has nothing to do with any of them.
I never served as an Assistant Rabbi, something which I think is a very good thing; I would likely have been awful. I would have been careful to avoid invading the Senior Rabbi's domain, of course, and I would have done what I was asked, but tzimtzum (reduction of one's presence) does not come naturally for me. But at one point I was asked by one of my avreichim for my thoughts regarding the Assistant Rabbi model, and here is what I told him.
To my mind, there isn't a single job description for an Assistant, any more than there is a single job description for a Rabbi (as we have discussed elsewhere on this blog). Each community has different needs, each Rabbi has different needs, and each Assistant has different skills.
Here, though, are three possible models, based on examples from the Torah:
1. Eliezer, servant of Avraham - Eliezer has no real voice of his own; he is meant to speak Avraham's words. Indeed, when he tries to improvise ("perhaps the woman I seek for Yitzchak won't want to come to Canaan") he is shot down. We don't see any special talents in him. Eliezer is assigned to take care of specific jobs, and he does them. [Gechazi, Elisha's servant, may also be of this model, but Gechazi proved untrustworthy.]
2. Yehoshua, student of Moshe - Yehoshua has positive traits as well as weaknesses, which are displayed in moments of crisis like the incident with the Spies and Eldad and Medad's prophecy. Yehoshua is given areas that are under his control, like war with Amalek, but he clearly answers to Moshe. It seems clear that any autonomy he owns could be withdrawn by Moshe at any moment.
3. Aharon, second to Moshe but also his peer - Aharon aids Moshe in conveying his message to the Jewish people; he is Moshe's navi, speaking on Moshe's behalf. In this sense, he is like Yehoshua. However, Aharon also has his own particular job as kohen gadol running the rituals of the Mishkan, and his own particular relationship with the community, largely independent of Moshe.
To my mind, the Eliezer model is unhealthy; if you apply for an Assistant Rabbi job and it sounds like that one, run the other way. It is unlikely that someone will go through rabbinical school just to become an Eliezer.
I could see the Yehoshua and Aharon models being healthy, in various circumstances. Perhaps a young rabbi could be a Yehoshua, and like Yehoshua he could evolve into someone who is ready to be a leading Rabbi. And the Aharon model sounds great - but it would require complete bi-lateral trust. Good luck...
There are probably more biblical models out there; what would you add?