Requests are not for getting material gifts from other players. They are only used to get help in a quest. The quest info will say "Ask" and this will bring up your neighbors list. Check the people you want to get the messages.
You can only receive 20 requests per day and if you try to request from 30+ people, it doesn't work.
Does it cost them anything? No! Your friend will see a message that you asked for Minstrels, or Exotic Seeds, or whatever. These items do NOT come from the player's inventory!! All they need to do is click the button to send it.
Note: even if you receive a request that is hours or days old, go ahead and click it. They might still need it to reach their total!