It depends on what you mean by flow - the movement of the gas (as shown by a flag waving in the air) or simply the movement/homogenization of the contents of both inside the chambers.
In the case of visible rushing of gas from one chamber into another, it's the total pressure gradient that matters, in the case of homogenizing contents, it's the partial pressure. Let's explain by example:
If Gas A and Gas B are at equal pressure on the left and right side, if you have a flag in the middle when you remove the obstruction you won't expect it to flap into either side. It would probably stay centered, but you would notice that some of Gas A starts to move towards Gas B's chamber and vice versa. Depending on the temperature and other factors, this may be a fast or slow process.
Now, if Gas A was at a lower pressure at the time of opening (because there was less gas A), you would see the flag flap to indicate that gas B was rushing into Gas A's chamber. The magnitude of the flow being proportional to what the absolute pressure difference was.
A good way to see this is to actually use water with two different colored dyes.