How do bound NPCs spawn?

Solution 1:

The bound NPCs spawn just like monsters: They appear in a radius around you, off-screen, and will despawn at some point.

If they are killed or despawn by themselves before you get to them, they will eventually spawn around you again.

All of them will only spawn while you are underground. Additionally:

• The Wizard only spawns in hard mode.
• The Mechanic only spawns in the Dungeon.
• The Goblin Tinkerer only spawns after you've fended off a goblin invasion.

The Goblin Tinkerer has pretty much no preferred hangouts. See this answer I gave here; basically, the most effective way to find him is to wander around aimlessly underground while chugging Hunter Potions (it reveals bound NPCs just like monsters).

Playing fullscreen at native resolution also helps.