The main problem with cutting boards and sanitation is that you, well, cut them. Those cuts produce channels where bacteria can hide and survive even a good cleaning. But the main worry is with wooden cutting boards, because of how they get cut (cuts on a wooden board
produce deeper more 'cavernous' channels where bacteria can hide) but also because they are harder to wash well and also that bacteria doesn't mind that sort of environment.
I assume, since you said dishwasher safe, that you are talking about those thin silicon boards or something similar. For these it's not a bad idea to use separate boards but it is probably not necessary. They don't tend to have good places for bacteria to hide in the first place and bacteria doesn't survive on them very well, either. Also you can blast them with extremely hot soapy water for an hour on a dishwasher. At any rate, it's incredibly unlikely any bacteria that survived would bounce from one board to the other so I wouldn't be worried about storing them together.
Finally, keep in mind that by far the biggest issue is with foods that are not cooked. So my biggest worry personally would be doing something like preparing a salad on a wooden cutting board that has also been used for cutting raw meat. It's just not worth the risk, even if I knew the board was cleaned as much as possible. I still use separate silicon boards, but that's more for caution and so I don't have to worry about whether the board was perfectly cleaned or not.