There's a lot of good advice already been posted......
I would suggest also wherever possible that when you get where you're going, park up away from the main slip if you can, and go on foot to have a good look at the area.
That way, you can see which will be the best approach for reversing a trailer, or if the end of the slipway has a three foot drop off, which you can't see, which side to put out your lines/ fenders, etc etc..
This will also allow you both to work out a plan of action.
Once you know what your going to do, I would get the boat set up ready to go into the water, i.e. fuel cans in place and all the things you are taking out, lines ready. Leave the engine up.
It works well for us to put a long line on the bow, and a line off whichever side the boat will be tied up to.
Reverse the trailer into the water, until the car wheels are nearly at the water, then stop. Make sure you can see the other person, and they are not behind the trailer, or anyone else for that matter.
Switch off the car engine, put the car in gear + handbrake, then release the tension on the boat winch, and allow the boat to slip off the trailer, controlling the winch.
Once it's floating, the person at the side can use the stern line to pull it clear, and you can follow up with the long bow line, and both tie off the boat where it will not block access for others.
Then simply pull the trailer out, and if available hose it down wit fresh water.
I would only put the engine down once you are totally happy it's deep enough, and would consider using paddles to move to a more suitable area for start up if required.
I would say the golden rules, are to look before you put the boat in, and do everything slowly, and don't allow others to rush you.
If you prepare well, before you launch, you can do it without even having to talk to each other much, it looks far better to those watching, and hopefully you can avoid any drama or embarrassment.
Hope it all goes well for you.