As a flash user i can say - that yes 1 to 1 and 1 to many players game is possible to make on flash. I see some games like this in social webs.
I do not make such a games but i make a simple chats on flash - the idea is the same:
You need to create a server (for example PHP) - and client flash.
How it works:
Server send messages to both flash and they send massages to it. After flash Read and calculate some action on its side it send massage to server and etc.
It is just a theory - in realization part it is not such simple
Just lets try take a simple game like chess without any optimizations.
We have 2 players. One make a move - it sends to server for example: we2e3 - the another flash reads it - and moves white object in e2 to 3 e and etc.
But for example - where is it possible to use in porn game ? I do not think that people will be interested to choose how there cherecters will fuck or etc
Yes my dear - i want to use Pose nubmer 5 - oh no i do not want it - only Pose 4/6/7 - ok let it be 7 and they se ies animation - new sex online game
Just a joke
In action games too - it is not so simple - yo have to send message as soon as you want your game be dinamical. Any changes = message. Any move = message.
For free servers i do not think that they let you to do it. 2 much use of there CPU. For example to make it on private server - but then you will need to not to shut down you PC
It is another solution for 2 player games to for example - peer to peer communications. You connect 2 players and they play like 2 player game on old schoool gaming platforms for example. The same like with server - but with out it
- so if you wil not need to pay for server - but you will notneed with server too if only 2 players in game ether.
---
The same game will work on smartfons and PC - anywhere whee flash does.May be it will even work if turn it too apc for smartphone. But i did not test it like that. But the main idea is - that it is hard to make as you already understand.
You need to use more then just a flash - so it takes time.
Then you need to make good content for game in flash - it takes time like to create just a same game on flash but 1.5-2 times more - for optimize it work with server.
And then the hell starts - you will need too optimize and test it. Remove mistakes - mistakes with "if one player have bad traffic" / "to much gamers" and etc.
So it will take i think 3-5 times more time then just make a ordinary game on flash.
Lets showon simple example for example chess:
To make a simple game like chess where 2 players play from one keyboard and PC - will take less then a week.
To make it play on the internet - 2-3 weeks. If the same 2 players will play it and only they.
To make more players - 1 -3 weeks more. And in future some time to fix possible bugs and work with server lugs politics and etc - if it become more players.
It is like not real number but i think it is near a real - may be real wil even take more time. And it is just for a simple game like chess.
For example some arcade games was made by i think near year time. And some of them were made more then one human. So it will take a lot of time.
And if make some thing like that it is good to make big content - it will take very many time.
Or just make it simple - with simple idea like chess and etc - or note to hard game algoritms where there are some step by step game play - for example like Heroes of Might and Magic and etc.
So for some thing like this it have to be some global idea and a lot of people who will do it. And as i understand people there do not like global ideas
- thats why i think some projects there fail. Lets name this communication problems