I think it would depend on a few factors.

Firstly: Location. If you live in NYC or Boston or Chicago or California it would be a whole heap easier to find someone than if you live in a small town.

Secondly: Financing. Do you have any form of health insurance? I don't know much about the US system (I'm assuming you are from the US because I'm usually right with that assumption, but if I'm wrong I'm sorry). I think that if you do have some form of health insurance then your insurance company should have a list of providers where you pay a fairly okay part charge and your insurance company picks up the rest. That could mean that you could indeed see someone in private practice.

Do you want treatment at a training institute? Um... I was a bit confused about that... If you live in one of the above mentioned places the psychoanalytic institutes sometimes offer sliding scale rates to people if their financial situation prohibits. I'm not 100% sure (I think times are changing) but I would think that the people training at one of those institutes would tend to be older (and typically PdD or MD or whatever in addition). So it isn't like they are 23 years old... And it isn't like they haven't had life experience... And it isn't like they haven't had experience with seeing clients... It is just the framework that they are new to...