We went VA from Manchester and were told they would collect our bags for us if we took our baggage tags to the ME checkin desk.
When you get off the plane at MCO you clear customs, then collect your bags. Follow the signs for a little way and you get to another conveyer belt which is staffed, at the foot of an escalator taking you up to ground level. You have to take the monorail at the top of the escalator to the other terminal to DME. They give you the option of popping your luggage on the conveyor belt or taking it with you. I always drop mine off as I go alone with my little girl and I cannot pull two full size cases plus hand luggage and get it up the escalator. However, if you are anxious to get to your resort and want to take your bags with you on ME, and you can manage your own bags I would suggest you take them with you.
you then take the monorail to the other terminal where you either take your tags to the ME check in with your Disney resort confirmation/vouchers or reservation no. This is if you want them to collect your bags. They will direct you to which lane to enter on the "Red Carpet" and you queue there for ME.
If you want to take your own bags, you wait at the carousel for them before checking in at ME counter and then take the with you to do so. To be honest, we take ours ourselves. BUT ... the reason I said to take your own bags with you if you can manage them from the arrival terminal is that they take anything up to 40 minutes to arrive at the other one so you could probably get an earlier bus if you had them with you via the monorail.
By the way, we stayed at ASM last time and *** was the first stop. Easy Peasy!!
I seriously don't mind tipping the driver as a few dollars to make life easier on a ££multi-k holiday is worth it for me. I am tired and crabbit when I land in my winter coat in 35 degree heat and I will pay for whatever helps!!

Also the ME drivers have always been great and I enjoy the unwind on the bus on the way there.
RE. new policy, I was told by Disney UK when booking for our upcoming trip that they will no longer be collecting the international travellers' bags but we still get to use the ME so you may want to check with whoever you booked through and confirm this if it is an issue.
Feel free to ask for any more info.