You can have my bag...

[Boozy]

But they aren't accepting your personal items into their care. They are simply saying that "We don't allow bags in the store."

What you choose to do with that bag is up to you. You can leave (as I do), or you can drop it at the front and take your chances.

[Lace Neil Singer]

That's a weird rule; round my neck of the woods, nowhere has a rule like that. True, some places like museums and concert halls have the right to search people's bags, but no stores have a "no bags allowed" rule.

[elsporko]

I don't have alot of stores taking my bags either, but it may be a big city/small town disparity. More crime in a city will mean stricter rules.

[Amanita]

The setup Plaidman described sounds like it makes sense- your stuff is kept in a secure spot where only employees can access it, and they have a means of making sure that the right person leaves with the right bag.
There's no freaking way I would "leave my bag outside" in any place, where there exists no means of securing it, and anyone can walk off with it. I remember hearing about one store where they made people leave their belongings in a large bin that was completely open, and anyone could walk up and take things from it. Needless to say, a lot of stuff got snatched from that bin!

And yes, around here they do say "You have to leave your bag with us." in places where that rule is enforced. So yes, they ARE asking you to hand over your property to them. If that's what they are doing, then yes, they should be liable.

[NorthernZel]

Um, over here stores who run a "no bag policy" most often have an area with lockers where one can put his/her bag(s). The lockers are normally placed next to the Service Desk so they're monitored, and I do think that if someone should succeed in breaking into one of the lockers without the Service Desk noticing, the store is liable.

[Lace Neil Singer]

I don't have alot of stores taking my bags either, but it may be a big city/small town disparity. More crime in a city will mean stricter rules.

Haven't noticed it in stores whenever I visit London. However, the bag searching I mentioned earlier is rife in museums, especially the free entry ones.