Many Sources. See your pediatrician for guidance. Most schools have testing avaiable if needed.A specialist can help with diagnosis and proper therapy, if this is indeed present.
Start with school. School are required on parent request to look for ld and then provide services if it is present. Go to your school districts website and look up special education. The procedure for seeking testing will be described there. If you don't feel testing was adequate, you can seek psychological testing elsewhere; your pediatrician may be able to make a referral.
Public/private. You can ask that this be done but do so in writing.Be sure to sign documents that allow the school to do so and share information with the testing source (there are strict laws covering student info).If the kid does not meet the schools criteria you can have this done privately with an educational psychologist. It can be expensive but often a way for you to qualify your kid for special services.
A few ways. If you suspect your child has a learning disability, you can ask your school to do an assessment. You can also ask your doctor for a referal to a developmental pediatrician for an assessment.