I've ridden a fixed road bike with trainers / flats without too much of a problem. Although it's not as easy to 'brake' via the pedals I never had any trouble keeping my feet on the pedals.
However...
The first time I rode my MTB after converting it to fixed wheel I put flat pedals on. My thinking was that flats would make it easier to just lift my feet off the pedals on the downhills and get off the bike in a hurry if I needed to. On the first dip in the track I came to the pedals just vanished from under me and then promptly hit me VERY hard on the back of the legs.
That was the last time I ever rode the MTB fixed wheel without clipless pedals.
I rode fixed for the first time yesterday. My new bike (Pompino) came and I rode it round the block. I'd put on flat pedals and had a couple of 'moments' where I tried to freewheel, without success. I think that if I had foot retention, my descending foot would be pulled down, which would remind me what was happening and left in a position where I could regain control, rather than being left hanging, waiting for the pedal to come back and scrape off some skin - which is what happened.
My conclusion from 5 minutes riding and reading this is that:
- if you know what you are doing, riding fixed without retention should be fine
- for a learner, retention probably helps to get the hang of it.
I hope to come back shortly with personal experience of the second point!