Mine are about 15- 20 feet apart and are taller than my rambler. When they grow into each other I'll prune them lightly so it looks attractive like an arbor- but naturally- thei'll frame the front door. Don't worry- be happy- I have another one in the back and I love them.

River Birch are large trees I have seen many planted singly at the corner of a house and have overgrown the spot. Planting together will be ok for several years, however they will eventually grow together. Too tall to prune

If the look you are going for is a grove of trees, consider buying 1 more. Three looks better than two. Otherwise space properly.
Ask the place where your are purchasing the trees. They should know proper spacing.

Mike

Here is a link that might be useful: river birch