You don't need to sign a contract necessarily - you can be on 'flow' with no contract, or simply through a rep.
In both of these cases though, a company/rep would give you product and then state "you are now sponsored". Could be replaced with "sure I'll sponsor you" or "I'd like to sponsor you."
The important part is that you would get more product when you break it/are done with it/next year's product comes out.
There is a very fine line between 'hooked up' and 'sponsored' though.
For instance, you see a Smith rep.
"Dude you're awesome. Want to wear some Smith stuff? I've got a sick pair of goggles right here."
This COULD be sponsorship, but without conformation you might not know. The rep could just be getting you 'hooked up' because he thinks your rad. This is akin to holding hands on a date... you're definitely not official yet, but things are looking up. You're now 'hooked up'.
If that rep were to see you again, invite you to a demo day, ask you to go to a contest they're sponsoring - and especially if they do the "Dude we gotta get you on some fresh product." - then you have a surefire sign that you're probably sponsored or very close to it.
My very first sponsor was simple. I had been riding on some product, but I was applying for a contest and I had to list sponsors. I asked my rep "So, do I put Head on here or not?" - He said yes and I was officially sponsored.
Of course, you are not pro until you have a contract where the company pays you to ride their product. The difference between sponsored and pro are massive and there are many, many, many steps in-between.