I assume your borders are polygons. Near tool returns 0 distance because your points are inside their own polygons. You can try the following:
1. Run the Generate Near Table tool, specifying 2 for the "Maximum number of closest matches". The resulting table should contain two rows for each of the points: one row with a Near_DIST 0 and the other with a non-zero Near_DIST value.
2. Use Select By Attribute to select Near_DIST > 0.
3. Use Join Field tool to join the selected table rows with the points; you should be able to get the NERA_FID and NEAR_DIST fields transferred to the points.
Hope this helps.
If you need to know what are the shared borders, that would require more analysis. Is that your case?