In the context of social media, the distinctions outlined in some of the replies here don't really apply / aren't observed, with terms direct message and private message being used interchangeably, though perhaps platform-dependently: e.g., Reddit has private messages while Twitter has direct messages – they are both private in nature (although multiple recipients and group "chats" are possible), but perhaps indirect given the architecture of these internet applications, or the internet in general if you want to be really technical about it, although they appear direct to sending and receiving parties.
I'd say private message (PM) is the term people normally default to, but some (younger?) people will often say DM me ("direct message me") regardless of the application. I don't want to investigate this in detail, but some insight may be gleaned from Google Trends – I'm purposely steering clear of corpora as I don't see any indication of year in iWeb or GloWbE, both being Web-based corpora – where I searched for the frequency of queries
how to direct message and
how to private message in the United Kingdom (here) and the United States (here).
In the Related queries boxes below the graphs you can see which term is used with which popular social networking service.