What exactly does participation entail? I snuck out to do a couple of quests just now, and there was a message from someone on my friends list to participate in a level 9 raid. I clicked yes for that, and nothing happened except the prompt went away. I then did two quests, and didn't get a demon summon after either of them.
Since there are some people who weren't around for the last event I'll try to give a detailed low down.
For the duration of the Demon Raid there's a display that reads "Souls" at the top of the screen and displays your souls as yellow orbs. You can have 3 at one time. Whenever you complete a quest there's a high chance you'll get a message that says "DEMON INCOMING!" and displays its level. When this happens the display at the top will flash and say "DEMON INCOMING!" and have a number in the top right showing the number of demon encounters. This will usually be 1.
Tap on this to display a list of demons you can fight. Each demon will have a timer on it. When you trigger a demon by doing an AP quest, you'll have 1 hour from that point to complete it. Unlike normal fights you're on a time limit. If you spend 1 soul, you have 90 seconds. If you spend 3, you have 200 seconds. Make your choice, then fight through a few waves of enemies to kill the boss. Afterwards you'll get a chest with a card inside. You also get demon coins for initiating the raid and defeating the demon.
Each time you defeat the demon it will increase one level. Killing higher level demons gives better rewards and you get certain rewards for clearing specific levels, most notably a 5 star card of the demon. The last demon wasn't very good so most people sold her, but this one is supposed to be good so hang onto it.
Eventually you'll reach a point where the demon's too strong to kill in the allotted time or kills your party. At this point the game will ask you if you want help from friends. It will show a random selection of people from your friends list and you can refresh to list to get a new selection. Once you confirm this they will see your demon in their list. If someone asks you for help you'll see theirs. At this point you're free to spend more souls to try and kill the demon yourself. Everyone who participates by spending at least one soul and fighting the demon gets demon coins. Whoever deals the final blow gets the chest drop.
At first people will probably be killing their demons themselves, but once people start reaching their limit more and more requests will go out. Last time a bunch of us gaffers banded together to get some people up to 80, even if they could only make it up to 50 themselves so don't underestimate the power of friendship!
So, in short what are the rewards?
Notably some 3 star weapons, a 5 star warrior demon, and lots of 1-3 star arcana cards. Each chest drop gives you character arcana, enchancers, or weapons. At higher levels the loot improves and you have a decent shot at 3 stars which can be used or sold for rings. Demon coins likewise can be spend for the same sort of arcana with a rare chance for the 5 star demon. If nothing else you'll probably get a good amount of fortune rings from selling 3 star arcana.
How do I get more souls?
You can have 3 souls max at any time. If you have below max, they will restore at a rate of 1 per 30 minutes. If you rank up this will restore all 3 souls. There are items called vigor fruit available from the store and sometimes as rewards for completing specific difficulty levels of the demon raid. I believe refreshing stamina with prysma or stamina fruit also restores all 3 souls. I'd recommend saving any items that refresh all souls until the higher difficulties where you'll burn multiple souls to kill one demon.
Stuff to be aware of
There were some bugs during the first raid. Most notably if 2 or more plays fought the demon at the same time and killed it, it would bug out and have 1 hp remaining. It would essentially become unkillable and a waste of everyone's time. If I recall correctly the other issue was completing an AP quest and getting another DEMON INCOMING! notification while you already had one of your own on the list would cause it to refresh its health. I don't know if these are still issues, but its something to watch out for.