Median assessments hours were 148 for applicants in the most recent APPIC survey. This certainly suggests many people are moving on without much in the way of assessment experience. Obviously standards will differ across settings but if that is the median for applicants I'm sure at least some people in that range are going to hospital settings.
I think most places want "some" experience but I don't get the impression many places expect much in the way of assessment experience unless you are applying for neuro, forensics, etc. Truthfully I don't know how they could with those numbers - I'm closer to 250 hours and still consider myself borderline inept with regards to assessment. Yeah, I've gotten some variety and have the basics down (e.g. PAI, WAIS). I'm confident I can pick up a manual and learn to administer/score just about anything. However, there is a world of difference between that and being able to know the best battery to administer to allow you to tease out a particular nuanced aspect of cognition. Its different training, for individuals working with different disorders, operating in what is really a very different world than the one I work in. I've gotten enough exposure to (hopefully) do some basic screening and know when to refer. To me, that seems reasonable for someone not planning on a neuropsych career but we'll see if internship sites agree when i apply this fall!