No matter how you spin it, if you want these two rockets to go in exactly opposite directions and meet up again, The point at which they meet would _not_ be partially sideways and partially "up" (away from the earth). at the point where they meet, they would be going 100% parallel to the ground, and 180degrees from each other (ie, direct head-on collision).
This is ignoring the fact that minimal stable orbital velocity (the minimum speed for your rockets to go 'around' the planet, is about 7.8km/s, or 17 thousand miles per hour. and because they're going in opposite directions, thats 34 thousand miles per hour of effective collision speed.
Now, if you're suggesting instead, a full 'escape' velocity from the earth that allows them to both enter parabolic trajectories, and meet at some arbitrary point in the distance, I hope you've thought of a way to allow your two spacecraft to survive an explosion that launches them from 0 m/s relative to each other to roughly 22km/s relative to each other (49 thousand mph), in some semi-instantaneous event. If we assume this 'push' from each other happens over 10seconds (this is generous), that's gaining 2.2km/s/s, which is 224G for 10 seconds.