or would it stand out in the open and expose the spy?

I knew it was a bad idea to ask a question here......n

I doubt it very much.

I don't think any critter has a stealth mode.  There's certainly no stealth skill you can train them in.

So the pet would be out in the open while the spy is running around stealthed.

Dunno if you set the pet on a target, and you'd still retain your ambush bonuses when you attack.

 no.  and it follows around while your stealthed, 

Yes.

As of the Game Update for Spies, Pets go into stealth whenever their master goes into stealth.

This applies for spies, jedies who use their cloak, and players who are cloaked by an allied smuggler.

Pets that go into stealth will go out of stealth if they are in combat.  If they are pulled into combat while in stealth, they will pull their master out of stealth too.  Its kinda buggy then, because AoE's can easily screw a spy/bm up.