There's definitely competition among the selective schools. Grammar schools simply see more applicants than they have places available for, so you can expect a high level of competition here. Some grammar schools see 3x-10x as many applicants as there are spots available.
This is one of the main reasons students and parents get a bit stressed, so it's key to take preparation for the exam serious. The admissions policies of the schools are generally listed on their website and evolve around the 11+ exam results. Is thus important to perform at your maximum potential as the pupils with the highest scores will receive an admission offer.
I would also recommend to apply to multiple schools, so that when you’re missing the boat for your favourite grammar school, there’s still a chance to get accepted by another grammar school. By stating your preference it doesn’t matter whether the school is within your catchment area or not. You might even have to do multiple 11+ exams when applying to multiple schools. The main take-away is that you don't want to end up in a scenario in which you applied to only one school and just didn't get in.
However, please keep in mind that when selecting multiple schools, your child might indeed also have to do multiple exams. On top of any school tours, interviews, discussions, and the terrible wait after the exam - it could get easily too much. So do select a few schools, but keep in mind that it become very hectic.
Why not give the 11+ courses a go? There're for free!