You'll first need to ensure that the points you want to join to have x,y fields which are populated. I'm guessing you already do, but Calculate Geometry Attributes—Help | ArcGIS Desktop will do this if not (POINT_X, POINT_Y).
Also ensure you have everything in the correct projection/coordinate system.
Spatial join normally only returns the count of matching features, and the attributes of the first feature it encounters.
However, the Field Mapping allows you to specify some very handy rules on how you handle multiple coincident/within a distance features.
In this case, you would probably want a field containing a delimited (comma separated for example) list of all the IDs of points within a distance from your other input features, and another field containing the XYs of those points.
A helpful blog (helped me several years ago) here Spatial Join’s hidden trick or how to transfer attribute values in a One to Many relationship | Esri... should give you an idea of the process (beware of max field length in your new fields!).
