you ran heroic dungeons to get badges, once you got enough badges (pretty sure they were the same cost or even more expensive than current tier raid gear) you bought them from the badge vendor. after making sure it is enchanted properly you send it off in the mail to your chosen alt (so same faction and realm like you said).
there were heirlooms for Shoulder, Weapon, and trinket slots on the vendors i believe, with a ring being a prize available from the fishing contest. the PvE heirlooms cost 40-65 badges, while the PvP heirlooms cost roughly 200-325 Stone Keeper’s Shards which drop off any Wrath instance as long as your faction has Wintergrasp