Of course. We appreciate that. But will they accept, or no?
It depends on the situation and in some cases yes. Each private albergue is (obviously) privately owned and there isn't a set of rules that covers every situation.
In one case I injured myself early in the morning, when I got to the next bar and stopped to rest I discovered that it was also an albergue and they had a pile of bags waiting for pick up.
I asked the albergue host if I could add my own bag and was told yes. In that case I had not stayed in that albergue that night and I hadn't even stayed in that town.
The bag pick up service doesn't call at every albergue, they only call in to pick up bags if someone has phoned them (or messaged them) the night before to request a pick up.
If you left your bag at an albergue where there were no other bags waiting and you hadn't specifically asked the bag company to pick up your bag from that albergue then you risk not having your bag with you that night.
The way that the service works is that the various bag transport companies distribute their own branded envelopes to the albergues. When you arrive at the albergue you pick up an envelope and contact the company saying that you want a pick up the next morning.
The next morning you write your name and phone number on the envelope along with the destination albergue. You put your payment into the envelope, seal it and attach it to your bag.
Later that morning the bag company calls in, picks up your bag and any others. Checks that you have paid the correct amount and then transports your bag to the destination albergue.
It is worth knowing that sometimes when they drop your bag off there is no one available to accept it and so there is a possibility that your bag will sit on the street until you turn up.
Here is a photo of a bag that was left outside on the street until the pilgrim turned up to claim it. You can see the blue, yellow and white envelope attached to the bag.
In general, albergues are not paid for managing bags and most will not take responsibility for anything that goes missing.
Some volunteer run albergues refuse to deal with bags at all. The bag above is outside one such albergue.
Some albergues will assist pilgrims by phoning in your pick up request for you but they are not obliged to do so and so if they do help you in this manner then I suggest that you be appreciative of that.