Hi shaela21,
I have two 5 gallon tanks, each with a betta. The tanks are arranged in an L-shape on the kitchen counter. The fish can see each other, if they are in the corners of the tanks that are close (about 3 inches) to the other tank. Male bettas like to flare at each other, but after awhile, mine got used to each other. I may catch them flaring once a week -- not very often.
At the hatchery I go to, there are many tanks full of different fish, all next to each other. One tank may have another tank on the left, on the right, and behind it. I don't think any of these fish are stressed from all the other fish. They get used to each other. At most pet stores, tanks are very close together, so they usually get used to other fish before you even get them.
I think you will be ok putting the tanks close together. Watch the fish closely when you first get them, as you would any tank, to make sure things are ok, but I think you'll be fine!