Our first "present" was a pork chop (uncooked) DH had left it on the side briefly, the cats got to it so he put it outside in the bin, they brought it back the next morning.
Second present was an out of date bag of mince. 
My cats are stupid.
Unfortunately since then we've had mice and birds, mostly dead. Believe me you'll feel terrible but they're better when they're dead!
The mouse took 2 days to catch, with a trap... I'd have rather the cats got it, and it was the 1 weekend DH was away.
The bird was easier to catch but it had been in the house for hours and was that traumatised I didn't expect it to live.
Just remember to give them lost of affection and praise.... or curse them but in a proud, pleasant tone of voice.... or they may bring you something bigger.
My friend was moving house once and was very upset, crying lots. Her cat brought her a bigger gift each day for a week... on the day she moved she found a hare one the kitchen doorstep.
To make matters worse her neighbour when she was saying goodbye said she was upset as the children had found an injured hare, brought it home and were keeping it in a rabbit hutch while it healed. But some hooligan had ripped the front off the rabbit hurt and they assumed from the blood killed the hare!
My friend didn't own up.
Neither of my cats wear collars, They've never kept them on for more than a couple of days, but as they've got older the presents have reduced in quantity, we rarely get them at all... or maybe they feel we don't deserve them now. :)