“Il se la pète” is quite a common colloquial idiom for an arrogant person.

Here are some equivalents, roughly sorted from the more polite to the more vulgar. Some have nuances, but all means more or less "He is acting without any fear of the public opinion, because he has a too high opinion of himself"

1. Il est très cavalier.

2. Il est sans-gène.

3. Il a du culot / Il est culotté.

4. C’est un cow-boy.

5. Il est gonflé.

6. Il a les chevilles enflées.

7. Il se la pète / c’est un péteux.

8. Il ne se sent plus péter.

9. Il ne se prend pas pour de la merde.

1,4 are more to qualify the brash way of acting of someone.

6,7,8, are more to qualify the over-confidence of someone, whithout the "try it on" notion

2,3,5 seems closer to what you are looking for. Note that there is a subtle difference since 3 can sometimes be used in a non-pejorative way, whereas 2 and 5 are always pejorative.