There seems to be relatively little information regarding either of these engines on the web. Anyone have an idea where I can find more info?
Each of those is two V12 engines coupled together.
The 606 is two 601s:
Two DB 601 F or G coupled to work on a single propeller shaft for use in early Heinkel He 177As - 2,700 PS (1,986 kW) at sea level with a mirror-imaged starboard component engine supercharger, and derided as "welded-together engines" by Reichsmarschall Hermann Göring in August 1942, from the problems they caused with engine fires in the He 177A during service from their inadequate installation design; such problems were not present in either the Heinkel He 119 and Messerschmitt Me 261 which also used them.—Wikipedia
The 610 is two 605s:
Two DB 605 "coupled" (geared together) as a "power system" (71.53L / 4364.8in3), to work on a single propeller shaft for use in Heinkel He 177, up to 2950 PS (2,909 hp) for take off, 2788 PS (2,750 hp) at 2,100 m. Mirror-imaged starboard component engine supercharger.—Wikipedia
The "engine difficulties" section of the Heinkel He 177 Wikipedia article discusses the story of both engines. The 606 was notorious for catching fire when installed on the He 177, and later it was replaced with the 610.
You can get more information by looking up 1) which aircraft they were installed on, or 2) the 601 and 605 base models.