My mother wants to open her own bank account because she believes I am stealing/misusing her money. She has already tried to close the bank account we share without contacting SS or the school district she worked for to change the deposit info. Her Alzheimer's and Vascular Dementia diagnosis was just done last month, but I have had my suspicions for over a year. 

In 2021 mom got scammed, by this guy who groomed her for over a year, for almost 3k. I switched her deposits to my account, and changed our phone number so he could no longer contact her. For the last 2 years I have been taking care of her because she can't go upstairs and that is where bathrooms and kitchen are. 
When I tell her we have no money, she doesn't believe it is because I am trying to get her debt cleared up and keep on top of the regular bills. 

I have no idea what to tell her when she keep asking me to take her to the bank so she can open her own account. 

Hi Holly. and welcome to the forum.    Do you have power of attorney?  Hopefully you do.  If not, you need to get it.  But also importantly, do you have a personal relationship with someone at the bank?  They can flag the account such that if she calls, they know not to follow her instructions (again, you will need power of attorney for that to be the case).  This happened to us--fortunately I was already a signatory on my partner's checking account even before I ever had to invoke the POA, and the branch bank manager knew us and knew to call me if my partner called with any iffy requests.  Similarly, the financial manager at her investment company knew us both and flagged her accounts electronically such that she could not complete investment trades by telephone or computer without checking with me.  We never had to invoke it, but there were a few close calls--in the few months before memory care she kept calling him saying she wanted to sell all her stocks and put everything in gold (!).  If you don't have relationships like that, it may hlep to make an effort to develop them.  

I know how easy it is for our LO's with dementia to focus on money, my partner did it too and in her heyday was very financially astute, which meant she continued to care about it.  But I would suggest any delaying tactics you can come up with (we'll go tomorrow, they are closed today, whatever) and DON't discuss with her that you "don't have any money."  she can no longer understand rational arguments.  My partner responded to lighthearted talk that I was functioning as her executive assistant; but fortunately this was not an area where she expressed any paranoia and did trust me, which made a big difference.  She would ask several times a day, every day, whether our cabin tenant had paid his rent.  

You may need to talk to her doc about medications for paranoia and agitation, if you haven't already.  Good luck.  

Discuss how to title the new account with the elder law attorney that drafts the DPOA to avoid impacting medicaid possibilities , general and gift tax issues , inheritance issues etc. If her SS is going into an account now that doesn't have her name on it-check with the lawyer on that too . 

  And SS has a representative procedure ,they don't accept POAs . Once you hold the DPOA and it is inforce- freezing her credit is a good idea.

I would stop combining  her money with yours.  By that I mean, have a joint account but only with her money.  And only using her money to pay for her expenses.  If you put your own money in the account, it’s considered either of yours by the bank, and by anyone she is legally liable to. 

I am on joint accounts with my parents.  My money does not go in the accounts and I only write checks or use  a debit card on those accounts to pay for their needs.  I keep receipts in a notebook. If it’s for groceries, I staple the grocery list to the receipt.  I write notes in that notebook if they asked me to get cash for them or order something on line 

The standard usage of "joint account" is this:

https://www.forbes.com/advisor/banking/should-you-have-a-joint-bank-account/

Co-owners, co-liability.  The lawyer should advise on the right titling.The right Social Security number to use etc.

I've been advised the account owner  should be the PWD, or their trust   with the DPOA as the only person allowed to access or manage . But not "own."

 Otherwise the PWD could sign away the assets, the DPOA could be held liable for something in their life - lawsuit, unpaid bill and the JOINT assets could be attached-- totally unfair to the PWD. If the PWD wants the assets split between their children and they die with money in an account that becomes one child's... it's just messy.

 Would Medicaid consider the joint titling as a monthly gift/partial gift to the DPOA of the SSI  or pension if there is right of survivorship in the account ?

There are many ways to get something done and a banker gives the easiest procedure for them,  but since the only certainty in life is death and taxes best to get things set up at the beginning  to not backfire on the back end.