Training spears, swords and axes are edgeless weapons created at a carpenter's workshop. Dwarves won't harm each other (or themselves) during combat drills, or sparring, so they are only useful when you have no metal weapon of the kind you'd like your dwarf to use later in combat (or for Danger rooms). They are useless in combat. So make sure that your soldiers have exchanged them for real weapons before sending them into combat, even if it means using a weapon they are totally unfamiliar with.
Soldiers sparring for long enough can grow attached to their tiny wooden training swords. Congratulations, you now have a legendary wooden stick.