As the person who's named in Mom's POA, what rights do I have?

No matter whose house it is, she is still entitled to privacy in her "home". This is known as "tenants rights". You could not even evict her without going through the legal eviction process.

The house is my mom's and I'm named on all of my mom's banking, etc. My son is also named in the POA and who is going to stay with my mom to care for her in my absence while im away.

POA's have a strict responsibility to protect all information and keep it confidential. A POA must carry out the wishes of the grantor (mom). If mom wants her to know this stuff, discuss it, but only with mom also present.
Your aunt (her sister) does need to know who is coming into the home, right down to ID and background check, for her own protection. Since it is her home, she can deny admission to anyone she does not like.