If your PTO does not have its own 501(c)3 designation, then the IRS considers it a business and will assess annual taxes on your earnings. To avoid this, some PTOs will fore-go having their own checking accounts and will run their funds through the accounts of the school or school district. One of the main drawbacks to this is that you may end up giving up control of how your PTO's funds are spent to the school or school district administrators.
Basically, your PTO has three options:
1 - pay taxes on your earnings each fiscal year, or
2 - run all funds through another 501(c)3 such as the school or school district, or
3 - get your 501(c)3 ASAP!
There are thousands and thousands of PTOs that operate every bit as independently (arguably moreso) than typical PTAs. So the short answer to your question is: yes, PTOs can run as separate entities.
Only caveat I'd say is that i can't speak to specific regs or preferences of your particular district, but those would presumably apply to any parent group. There are districts that prefer a closer watch of PTO finances. But it doesn't sound like that is what you're talking about.
Generally, PTOs incorporate themselves with the state and register themselves with the IRS to become 501(c)(3) organizations. You can set up such an organization and control its funds; it doesn't need to go through the school books. Just follow the IRS rules.
On the other hand, if you don't play nice with the school district, they can probably keep you out of the school. I'd follow that rumor back up the grapevine and see what's up.
We have recently held our meeting and are on our way to dissolving our PTA. I "heard" through the grapevine that the PTO funds needs to go through our school books. HUH?!?! Is anybody out there working as a separate entity from the school or are you letting your funds go through the school??? I feel I have been thorough on my research on becoming a PTO but wanted to make sure Im not missing something.