My DS is in a grammar school; the local grammar schools take less than 10% of each year group. He did not have a tutor. He did learn the verbal reasoning types at home and I made sure he knew all the Maths required.
You can only get a child through the test, either by working at home or by paying a tutor, if they have a reasonable level of ability to begin with. For many of the question types you need a wide vocabulary; that has to built up over the child's life, so you can't teach it in the run up to the test. So teaching at home is more about methods and building up speed.
People on MN will always bring out this myth that the relatively less academic child only got into grammar school because they were tutored and will struggle once they are there. Grammar schools usually have some setting and the teachers, like all teachers, know how to cater for a range of abilities; it is just a different range of abilities than would exist in a Comprehensive school.
You can look at the Sutton Trust for the research into outcomes; it is very accessible and not too academic in tone. The research claims that the most academic children at 11 do not benefit from attending a grammar school by GCSE results unless they are from a low income family; they would achieve the same GCSE results in a Comprehensive. The children who do benefit from grammar schools are children from low income families and children who are in the lower ability range within the grammar intake when they join the school. At GCSE, these children significantly outperform Comprehensive school children who had a similar ability level at 11.
In other words, if you are from a low income family or your child is bright but not the brightest, they will benefit more than other children from attending a grammar school, and you should prepare your child for the test. Look at the 11 plus forum for advice on how to do it.