1. create a route (create route) from your stream centerline using begin and end measures of 0 and the length of the river centerline.
2. locate the xy points along the route (locate features along route) to make an event table of the route reference coordinates and offsets. Offsets should be negative and positive for right and left and there is an option to set which is right and left.
3. To review the coordinates make a route event layer using the linear referenced route coordinates and the route reference.
There are several options to each of these tools so if you find something doesn't work how you expected, try another option.