My three DCs are/were at a grammar school (super selective). They were not tutored. I went through the practice papers with them to get them up to speed etc. You can buy the papers from W H Smiths or go on any of the eleven plus websites. If they are bright they will understand the questions, it is just a matter of showing them the quickest way to get to the answer, writing fast, not dithering, leaving questions they don't know and going back at end if they have time etc, that sort of thing.
I would say if they pass the test then they will be certainly able to do the academic work once they are there. It is only the same syllabus as at any other state school!
Which part of the country are you in? Some grammar schools take the top 25%, others the top 2% so obviously some are much harder to get into than others.
State primaries, certainly in my area, are generally not particularly supportive of selective education so will not help/advise you in any way, I am afraid.
My DCs are clever but lazy so it is hard to say whether would thrive at a comprehensive or not. They may do (being top of the class rather than mediocre/bottom at a grammar school) but on the other hand might decide not to do any work and be bottom of the class at a comp. it is hard to say. All I would say is at a grammar school they are actually forced encouraged to work harder than they might actually want to do and so it takes the pressure off you to a certain degree to keep on at them. Lots of people will disagree with this but all I want(ed) was for mine to get the requisite qualifications to give them a choice later on in life. I do sometimes wonder if they would have been better off at a comprehensive school, who knows?
Certainly, one of mine would have done absolutely no work at all, given the choice. Now he is mightly (?sp)grateful he has a set of decent GCSEs/A Levels and soon a degree ( although not much good though now in the present climate...)