I think it kind of depends on the item.

Is it cleats for baseball?- There doesn't need to be 2 full set of sports gear IMO

Is it a winter coat? - Toss up. I think both parents should have a coat but it wouldn't be a hill to die on.

School related things? Computer,, sure. Paper and pencils- no, BM needs to have those herself.

Clothes in general- BM should have her own

Toys and games- with 50/50 they should stay at the house where the parent bought them.

Agree with the above, depends on what it is. After a while though, if the skids need something from your home that they knew they needed and forgot or just left it and now want DH (NOT YOU!) to bring it to them then no, not your issue. At some point they have to learn resonsibility. OSD did this with DH for a while, I do think it was to make sure he'd jump for her, causing him often a 2 hour turn around to bring her half done report to her at BMs and part of a lost work day but he did eventually stop doing it. And she stopped forgetting things.

I guess it depends on what it is.  We have 50/50 (2-2-5-5) and for the most part the skids have what they need at both homes.  Some things go back and forth - we don't worry about what clothes skids wear on an exchange day,  so at any given point there are BM clothes at our house and clothes from our house at hers. We do not allow their toys or tablets to go between homes. But there are some things we do "share."  Skids have ski equipment at BMs that she bought.  If we take them skiing, she allows the use of that equipment.  We allow SD8's dirt bike to go to BM's house if SD asks, as BM has acres of land for riding.  There are certain big ticket items that it doesn't make sense to buy twice.  Of course,  if either bio parent wasn't responsible, or didn't return the "borrowed" item, this approach wouldn't work. I also wouldn't like it if the borrowed items were things BM could easily buy herself, as opposed to big ticket items, or if there was no reciprocity. 

I agree with JMTB.  It depends on what they are asking to take with them... I would also say that it would depend whether the other household is generally reciprocal when it comes to the kid's items from their home.

Special use hobby equipment... I can see letting the kid take it to the other home occasionally.. or like a team sport on an as needed basis.  The kid shouldn't have to break in two baseball gloves.. etc.. Both parents shouldn't have to buy team uniforms when games happen during visitation with both.  

Even clothing items I might not have a huge issue with a kid wearing a favorite item to the other home.. as long as it didn't turn into a one way street where the child ends up taking all "nice" clothes to the other home.  

Toys/books/games... generally should stay with one home but I can see the child wanting to take their handheld game with them that they got for christmas... or a personal Ipad or something like that.

School supplies.. I would think that mostly a parent would buy for their home.. but that special graphing calculator?  why should both parents buy one? seems overkill and that kind of thing should be able to follow the child.

What kind of things are you talking about OP? 

As already stated by all the other responses it depend on the item. If it's a big ticket item like an expensive electronic that you and DH paid for, no it stays at your house.
If it's a favorite shirt of pair of shoes, then no that's not a hill to die on...or maybe for you it is, I don't know.
If BM is calling needing school supplies, or clothes or food items, no she needs to supply her own, that's what CS is for as you stated.

I am a minimalist so do not believe in using our planets resources unnecssarily. As long as the skids can remember to bring them back to use when at your place this would work for me. It will teach the skids to be organised and plan ahead.

This is my hill to die on! Because of the amount of CS that DH pays, all of SS's clothing and other needs fall on me. Yea I could not buy it but it's not his fault this mom is a selfish greedy wench. However, anything I buy stays here. I make sure he goes back in what he came in. I had to talk to him a couple of weeks ago because he keeps trying to sneak things over there that if he takes everything that fits for things that are too small he's just gonna be SOL and I won't be buying him anything else. 

A few years ago we bought SS's Nintendo DS's and would let them take them back and forth. A few months later, when SS's didn't take them to BM's house one time my DH got a nasty voicemail that WE forgot to send their DS's back to her house and needed to drive over there and drop them off immeditelty. My DH was considering it until I was like uh wait a minute, we bought them those, why do they have to go to her house at all?

We'd also get nasty emails and calls about wanting all  her clothes back, which we'd send back but then get nothing of ours back in return and she never realised how that didn't make sense at all. SS also started getting into name brand clothing and I was sick of spending $100 on a sweater to never see again once it went there.

So that's when we decided that nothing goes back and forth and what they come in, they go home in. If it were a different situation and we were dealing with a sane person I'd feel differently but unfortunetly it has to be this way to lessen the drama.