We still use an in-house help desk, but as of late they seem to be more of a ticket dispatch system than a real help desk. The issue that I see is that we're hiring people who aren't even ready for a helpdesk job, and who don't have the aptitude or aspiration to something greater. These people are content to answer the phone and fire off tickets (quite often to the wrong team) over and over each day. Yes, they do password resets / unlocks, but that's really about it. Conversely, we've got people employed as desktop techs that really need to take a step back to the helpdesk and learn the basics.
I've worked for a company that outsourced after hours help desk, and it was extremely awful. Between user complaints about the technician not speaking English well to tickets that made absolutely no sense, it was a bad choice from the start. Management saw a way to save a few bucks and jumped on it, but the company as a whole suffered for that decision. After a few years, it went full circle back to an in-house help desk, and they made sure to hire the right people to staff it. User and upper level support job satisfaction increased significantly, which resulted in reduced turnover... which saved the company money.