Se creen los dueños de la calle.
 

This use of creer confuses me. I would say, "creen que son los dueños de la calle." Would my use be wrong?

creerse = believe, think (consider oneself)
creer = think, believe (suppose)

The pronominal form is correct, as far as I understand the sentence you wrote in the title.

They think they own (consider themselves the owners) the street.

I agree with Rusty.
"Creerse", in this kind of context, means to assume something. It's usually used to express indignation about someone's behavior. "Se" is mostly used as an emphasis, I think.

- El jefe nos ha estado insultando todo el día. ¿Quién se cree que es?
The boss has been insulting us all day. Who does he think he is?
- ¿Por qué no dejas que te ayudemos? ¿Te crees la mujer maravilla, que todo lo puede?
Why don't you let us help you? Do you think you are Wonder Woman, who can do everything?
- ¿Quién te crees que eres para llegar a estas horas?
Who do you think you are to arrive so late?
- Juan se cree la gran cosa porque salió en la televisión.
Juan thinks he's a big deal because he was on TV.

Thanks both of you. It's a real departure from English to use creer without ser. In English, it's always to believe yourself to be...

I think "ser" is implicit in the sentences.
You can say "¿te crees que eres la mujer maravilla?" or "se cree que es la gran cosa", but we drop it to make a shorter sentence, I guess. :)

Not really. "He thinks he's God's gift to women", "he thinks he's the bees knees", etc.

See also https://en.wikipedia.org/wiki/Zero_copula

Yes, but wouldn't it be he thinks himself God's gift to women be the way it would be constructed in Spanish se cree un regalo de Dios--thus leaving out the verb to be altogether?

Mea culpa. I didn't mentally process the 's as is.

It took awhile for me to realize that it could be used the same way in English:They think themselves the owners of the street.