It depends on the context. For the phrase like "social problems," the first one is commonly used. If you want to say something like "she is social," the latter is better.

Oops! Is the word in question "domatagohi" in the "social" context? In that case, I'm not sure. I'm sorry.

Breaking the word down and looking up the parts seemed promising. "Doma" yields "earthen floor", or perhaps the "pit" audience area of an old-style kabuki theatre (slightly different from the Globe, these were earthen floored box seats, rather than simply a big open space for the crowds). "Tagoshi" means a type of palanquin, with two people in front and two in back to carry it. The two terms together don't seem to make much sense, however, but it is all I can find to go on. Even my big honking Shogakukan Kokugo Daijiten (close to a Japanese equivalent of the OED) doesn't list "domatagoshi".

HTH, and good luck!