Four layer PCB, leaks current on edges
If there is exposed copper at the edges, then the board was designed improperly.

Almost certainly whatever software you are using to design this board has a parameter that controls how close edges copper is allowed.  Verify the setting, and of course make sure the software is actually checking against this rule.

The software may have not known that the board was intended to be broken apart.  In that case, it didn't know to apply the rule.  Either fix the setup so the software knows, or check this manually.  Either way, it's your responsibility to make sure copper stops some distance from all final edges.  25 mils is usually conservative for edges the board house will make.  Break-away edges will have more positional slop, and can affect the board a little distance in.  For those, you should use a larger keep-away distance.

Routing a ground plane right across a break away edge is definitely a bad idea.