This topic has been discussed to death, but I'll give you my take on it.
The hardest 2 specialties to enter to is OMFS and ortho, so I'm going to focus on those. other specialties are more lenient in their expectations. But for these 2, you would usually need to be top 10% of your class if you come from a state school.
If you're going for OMFS, then you would have to write an additional exam, the NBME CBSE. This solves the problem of P/F part 1 dental boards. This test is designed for medical students, so dental schools who have their students complete their first 2 years with the medical school will have an advantage in this test, but with a USMLE review book (First Aid), that deficiency can be made up if the student is willing enough.
If you're going for ortho, most of the schools want you to write the GRE. You would also need to submit your undergrad transcript and your DAT score.
Schools don't want to penalize students for going to a state school, but they do realize that the Ivy league tends to attract higher calibre students on average. Harvard has a small class of 35, and they train their students to be leaders of dentistry later on. Most specialize in whatever they want. Columbia ranks kids in thirds, and if you're top third, you can pretty much specialize in whatever you want. If you don't, then it will be a tougher hill to climb. For Penn, they rank the top 10 students, and everyone else is ranked 11th. 17 students matched into ortho last year, and only 1 of them is in the top 10.
But if you're cream of the crop, you're cream of the crop. If you can make it to an Ivy, then you'd have no problem making top 10% of your class. But general rule is, if you want it bad enough, you'll get it. A lot of people get burned out after 4 years of dental school and just wanna work and make money. Some have other obligations, such as family, and would rather want to support their family instead of going further into debt.
Stats do show that ivy league/UCLA/UCSF tends to send more kids to speciality programs, but it is because they accept on average, a class that is of higher calibre. in the end, i'm saying that the individual will decide if they will get into speciality programs, not the name of the school.
This is not like in Fortune 500 companies, where they only hire HBS and Wharton students.