It does send the message and number to your carrier (and I think Apple as well). With AT&T, you can also manually forward the messages to the short code 7726 (it spells SPAM), but that functionality is integrated into iOS and Android so you don't need to manually do that.

https://about.att.com/pages/cyberaware/ni/blog/forward_7726

AT&T does say that they take action based on the reports, but I think it can be difficult because spammers can rotate through numbers pretty quickly and they don't want to block things based on a few reports.

For example, lots of people report marketing email as SPAM, but Google can't just block emails from the Gap that are complying with all the regulations around bulk email and which most receivers aren't reporting as SPAM.

But yes, those reports do go somewhere and they do try to take some action based on the reports. However, SPAM is a hard problem, especially for a communications medium that's meant to be near-instant.