Adults are responsible for maintaining their own property. The clothes wearer is responsible for checking the pockets. The washer might check, but in that case the wearer owes the washer a thank you.
Both - but for different reasons.
The clothes wearer is responsible for checking pockets chiefly to prevent the accidental loss or damage of valuables or delicate items, but also, to avoid inconveniencing the laundry doer.
The laundry doer is responsible for checking pockets chiefly to prevent damage to the machine, or contamination of the load by objects such as tissues or items containing dyes and inks, but also, as a safety net against the absent-mindedness of the clothes wearer, and finally to take advantage of finders-keepers on small amounts of cash, candy, etc.
The washer emptying out the pockets is also how my mom found out my sister smoked pot.
Whose! Whose whose whose!
Laundry-doer, that’s whose job it is. Though in my case I’m both, so I’m not sure my opinion counts.
What she said. I tend to change clothes a lot (work clothes to house sweats to outdoor clothes, back to house sweats) and sometimes forget to empty pockets on the premise that I may very well pick up that same pair of jeans or that sweatshirt to put back on. The NajaHub, who does all the laundry and will sometimes collect clothing to wash unannounced, has learned to check my pockets in order to avoid sending pens (“Hi, I’m NajaNivea and I’m a compulsive pen hoarder…” “Hi, Nivea!”), pieces of raw meat, and the occasional pack of cigarettes through the washing machine.
The man’s a saint, I tells ya.
ETA:
What s/he said!
At our house, the laundry guy gets to keep any money found in the washer afterwards. I usually do the laundry. 
I never keep anything in my pockets and neither does my daughter, so it’s not much of an issue when I do laundry. My roommate, on the other hand, NEVER checks his pockets so I always check the washer and dryer before I do my laundry. There is, without fail, at least some change in the bottom of the washer, which, at my roommate’s insistence, goes in my daughter’s piggy bank. I have also found folding money (gave that back), a soggy pack of gum, and three lighters in the washer, and tons of shredded receipts in the lint trap. Oh, and another shredded pack of gum in the dryer, which left sticky residue inside.
He also washes all his clothes, whites, darks, whatever, all in one load, but (apart from a mild feeling of horror) that really doesn’t affect me.
Oh yeah, I vote clothes wearer, for sure.
If I had a dollar for every lipgloss I’ve accidentally washed and dried…well, I’d have enough to buy a new lipgloss, anyway. But not to replace the clothes the lipglosses ruined. He never leaves anything in his pockets except maybe cash. So I have to check my own pants as I put them in the washer, but don’t check his. Then I keep whatever he leaves money-wise. 
Whatever I find when doing laundry I keep. I’ve gotten driver’s permit, the obvious money (I have a piggy bank in the laundry room just for this purpose), cells, pagers, pens, and the security badges for husband’s work. I used to have quite the Hotwheels collection too. The important stuff I sell back for extra chores. The Hotwheels hung out on the window sill until parking became cramped. I don’t get many items anymore because they have learned. I still get money, which still goes into the piggy bank for my annual trip to Vegas. With an absented minded husband and teen, last year I pocketed $123.18 - most in quarters.
They both are responsible, the wearer however has little incentive to do so.
Is it settled?
I’ve washed 4 ink pens (they don’t leak in the wash). I dried 1 ink pen (it leaked in the dryer). I blame the owner of the clothes (me). From a responsibility point of view the blame goes to the owner of the clothes but the reality of ruining clothes is a budgetary issue. ruining a whole load of laundry is EXPENSIVE.
The wearer.
Considering I only have to wash my daughter’s and my clothes, it’s usually not a big deal. I do remind her to check her pockets and that I keep what she leaves.
When her dad was here it was always a bonus when he’d do his laundry as I almost always came out a few bucks ahead - he never checked his pockets or the washer.
And I am guilty myself - I’ve washed more lighters than I can count. Usually do not realize it until I pull the pieces from the dryer.
I believe the responsibility lies with whoever decides that a given item of clothing is now ‘laundry’. If your system is one in which only clothing in a hamper is considered laundry, then whoever puts an item in the hamper is responsible for checking the pockets. On the other hand, if your system is one which the laundry doer collects items from the around the house/bedroom, then they are responsible for ensuring that any items they pick up off the floor/a chair/the hooks on the bathroom door are clear of pocket contents.
Redundant checks are always good, so as the laundry doer, it’s a great idea to check, but if something gets missed and ruined, and you got it out of the hamper in the first place, ain’t your fault.
Both. Redundancy is good. The last thing I had in my pocket was a baggie of cat stool, so this is very much on my mind.
Clothes wearer. I do the laundry, and it really pisses me off when Ms. Malienation leaves some sorta paper thing in a pocket and it shreds all over every thing, necessitating an unusually long no-heat drier cycle to clean all of the fuzz off of everything. Grrrrrr…
On the other hand, if she leaves money in the pockets that’s mine to spend as I please. On drugs and prostitutes, if I so desire. So there.
Along with most respondents, I go with the concept of shred responsibility.
I always go through the pockets when I am doing the laundry out of a sense of self-preservation to avoid someone else’s gum/pen/junk ruining another person’s clothes (or the washer or dryer).
(I keep all proceeds.)
The only time I could see being mad at the washer would be if the washer grabbed extra clothing that was not in the hamper and then failed to check the pockets. If I have not put it in the hamper, one may not assume I want it washed. (Of course, I have been the launderer for nearly every family load for 20+ years, and I do not grab random clothing without checking the pockets, so it is a hypothetical situation.)
shared responsibilty
Shred is what happens to tissue that gets left in the pockets.
The best answer! Years ago my younger brother (now departed) and I lived together as adults. I always did the laundry cause, well, not to talk ill of the dead, but lil’ bro was a slob who’d wear dirty clothes rather than do laundry. Anywho… I always checked pockets and stuff cause I’m compulsive like that. I still do that, even my own clothes, even though I empty all pockets before putting clothes in the hamper. Once I found a twenty in bro’s pants pockets. Bein’ the nice person that I am, I gave it back to him with the warning that from now on, if I’m doin’ the laundry, I keep what I find. So I did. And yes, at times I’d find a ten or twenty in his pockets and kept the money.
Another vote for both.