I'm about to receive equity in a company. Let's say I get 10% ownership. Does that mean that if the company makes $100 in profit every month, they have to give me $10?
No. Some profits could (and should) go to retained earnings.
You will only received income when it is distributed to shareholders. It happens when the owners decide it should happen.
Keep in mind that in the case you cite, $100 profit is made and held as retained earnings, the company's value increases by that $100. Your net worth would increase by $10 as you own a more valuable company.
With privately held companies things can be a bit dicey depending upon the ownership.