Do board game publishers design games 'in house', or do they merely publish games by external designers?
There's publishers who only have in-house staff, there's ones who farm projects out to freelancers, and there's ones who buy games off designers.

Some companies only use one of these models. Some companies use two or all three of these models.

E.g.,  


Buying Games from Designers: Richard Garfield went to Wizards of the Coast to try and sell them a complete game he'd made called Robo Rally.  
Freelancing: Wizards of the Coast said they were looking for a card game people could play between events at conventions, and if he could come up with one they liked they would buy Robo Rally as well. He came up with Magic: The Gathering.  
In-house staff: Richard Garfield was eventually hired by Wizards of the Coast and later came up with Netrunner while employed by them.  


So there's one person, who has designed games published by one company, under all three models.