its situational but also massively dependent upon what you have avilable to you.
I raid lead but i rely a lot on my officers and particularly my healer officers to sort healing assignments.
Each boss fight is different and as such rather than just running through an entire instance with one dedicated healer on one tank..the assignments change.
However, in general we tend to have 5 or 6 healers in a 25 man raid. Our typical set up on healers is resto shammy, resto druid, pally X 2, prst x2. (we have in one or two cases run 25 man raids with 4 pallies and 1 prst).
In this setup i would generally assign the two pallies a tank each and ask the druid to hot the tanks as well. raid healing is then assigned to shammy, druid and prsts.
Invariably the prsts also drop heals like PoM on the tanks and CoH centresd on the tanks for the melee classes. And ofc..guardian spirit...
It does ofc vary dependent upon where you are.. and the level of your guild in relation to the instance...our guild are now doing ToGC and are 1 shotting normal ToC with 4 healers.
I guess the answer to the question "who do i assign to heal me"..is Everyone. Be reactive, work as a healing team and understand whats going on around you.
We operate a seperate healers chat channel and all players are in vent.. that way the healers can communicate faster if they die/stunned and the tank priority healer is incapacitated.
Some classes are suited to raid wide heals others to spot healing assignments.
All can do the job and all should if needed. All have various benefits to offer and combinations of all are used.