In FFXI:

-You obtain a "Cipher" through various means and trade it to an NPC that gives you the Trust spell for "X" character.
-You cast the spell in the overworld/dungeons and the Trust appears to fight with you.
-How many trusts you can summon at once are limited by your progress in one of the storylines.
-Some NPCs have sidequests associated with them that unlock their trusts, while others are found throughout the main story.
-Login campaigns give you free points to purchase other Ciphers for all trusts not included above.

In FFXIV:

-May or may not be limited to Shadowbringers story dungeons.
-Will probably only be available within the dungeon.
-Will probably follow you around similar to Squadron members, and attack on command.
-Unlike FFXI you might not be able to be that creative with your trusts, since in FFXI many jobs could fill different roles, so you might be limited to selecting 1 tank, 2 dps, or 1 healer.
-Trusts so far seem to be based off the Scions, but more might become available later through other means.
-Will replace playing with other players if you choose, but you can still play through the dungeons with others if you wish.