Do I need two signatures if a check is made out to two people?

The way the check is made out is probably the way that the account was set up- as a custodial savings account. With the payee line the way it is, the check is only negotiable for the "custodian"- your mom. You can try to get the check reissued in just your name but the easiest thing would be for your mom to just negotiate the check at her bank and then write you a personal check in just your name. She can try signing it over to you but most banks are starting to not negotiate 3rd party checks.