If you don't need the data to be accessible as a Feature Service, you could create a map image layer that displays the data with offsets.
Add the move effect to your symbols and enable symbol property connections.
Have your X and Y offsets defined by the expression (random()-0.5)*50 or whatever distance seems reasonable to you.
Here's a screenshot of a points layer being offset by that expression. Original points in small black, randomly moved points in red.
- Josh Carlson
Kendall County GIS