Do researchers do their own statistics?
Both scenarios are quite common, and commonality varies by discipline - but I can tell you for certain both happen enough that you should not be surprised by either case. There is also somewhat of a hybrid approach, where many researchers do their own statistics (or have a favorite collaborator who does it for them as part of their contribution) but they may consult with a statistician for some additional guidance or advice, formally or informally. If I were forced to guess, I'd say that in most social science fields I'm familiar with (many with connections to Psychology) I'd be somewhat surprised if less than 75% of statistical analysis was handled by non-statisticians. 

In practice, it tends to vary by individual study as well, and will change over time with any given researcher or team. Some people do certain kinds of stats only (ANOVA for everything, yay!), and if they can't do that they won't do any - or they just do an ANOVA anyway. Some people do no stats of any kind, ever. Some people develop a quantitative focus and start doing their own stats, and then become known as "the stats person" and end up becoming involved as a collaborator for others with a focus on research design and analysis, even as they continue with non-stats tasks (coming up with their own research ideas, running their own studies, etc.). Some people end up running into problems they realize don't fit the stats they are comfortable with, and work together with an outside person (consultant, statistician, etc.) to guide them or handle the analysis, but only for that problem (ex: they realized they needed a survival analysis but hadn't done that before, so they wanted to be sure they were going about it reasonably but then did the actual analysis mostly themselves).

So, in a nutshell: researchers of all kinds do stats always, sometimes, and never, and the mix varies heavily by discipline, problem type, and weather.