Mom is displaying inappropriate behaviors. Should I talk with her or should I let the owner/aides at Residential Living Facility address?

You can't change her. I went thru this with my mom and now my husband. Mom was in rehab then moved to MC because she kept trying to "escape" and thought she owned the facility. Her memory went downhill so fast and she sounded just like your mom. We tried home visits but, she tried to shoot them. We took the guns away so, she started yelling at them and EVERY DAY she would fall. That's how she ended up in rehab to begin with.
Let the staff handle her. I had to switch mom from her regular dr to the facilities dr so she wouldn't try to get out of my car during the drive to and from the dr since she kept threatening to kill herself.
I will remember you in my prayers. Let the staff handle her and let some of the responsibility slide off of your shoulders.

Imho, let the facility staff handle this. If you say anything to your mom, it may fall "on deaf ears."

There is no "reasoning" with dementia.  You can talk until you're blue in the face, but she probably won't even remember the conversation.  Unfortunately, inappropriate behavior is part of the ugly world of dementia.  There might be a medication tweak that might help.  Mention it to her doctor if her behavior escalates.

Let the facility your mom lives in handle her new behavior. Speaking to her about it directly will do no good. She's not the first one they've seen and certainly won't be the last. I certainly know what you're going through.
I've been in senior homecare for a long time. I had a client with LBD and some mobility issues on my service for years. Her symptoms stayed pretty stable for a long time. Mild then moderate. Nothing I couldn't handle on my own. I regularly took her to doctor's appointments, shopping, for lunches, and to visit friends in the area she lived in.
The mobility issues worsened quickly. Much faster than the dementia at first. So I used a wheelchair for her. It was harder but still manageable.
Then very suddenly the dementia got worse. That's when I could not manage her on my own anymore and she became homebound and quickly bedbound too. She still had to go to doctor's appointments (even though they did nothing that a visiting nurse couldn't have easily taken care of) otherwise they would not refill her prescriptions.
She had to start going by ambulance to doctor's appointments. Long before this point the family should have put her in a nursing home. She needed a higher level of care than one caregiver could provide and she lived alone. Her family's priority was protecting the money and assets from the greedy vice-grip of a nursing home. The poor woman suffered from lack of care and this wasn't my fault or the overnight caregiver's. We performed our service with distinction. She became totally bedbound because one person couldn't manage her anymore. She also developed pressure sores. The family ended up getting rid of both of us and got her an inexperienced, incompetent fool to live there that was cheaper. So her suffering increased a tenfold then.
Back to your mom though.
She's at the point where you can't take her out in public anymore on your own. Definitely talk to the administration at her facility about what assistance is available for such events as getting her to doctor appointment. When the rude comments and inappropriate behaviors start that is when you stop bringing her anywhere on your own. It's also time to start looking at nursing homes for her. Sooner rather than later because the dementia will not improve. It can progress very quickly and she will be beyond the care the AL can provide. Better safe than sorry. Find a nursing home now for her.

You're not going to talk a dementia patient into doing anything, sorry.

The staff at her facility have doubtlessly dealt with this before, many times; let them handle it.

Unfortunately talking to your mother isn't going to help because her filters are broken and she can't repair them, nor can you, nor can the facility. All you can do when you are out together is to try and distract or redirect her attention.

As far as the facility calling you, it should be to keep you up-to-date on what's happening with your mother. They are being paid to care for your mother, which includes her inappropriate behavior.

Let the staff handle it. If she doesn't even realize that your children are grown, she is not going to remember you trying to set her straight on how to act. Her broken brain will aggravate you - it's her reality.

Sounds like her dementia is escalating. She obviously has lost her "social filter;" whatever she thinks she says and doesn't have a care about what others think. She may also have some hearing issues if she is talking loudly. As for the bossy behavior, that may be part of her personality and would be hard to change. If staff can keep her attention diverted to something other than directing others, that might be best. If not, she may benefit from a mild medication to help her relax a little more.

I would allow the ALF to address. I first would speak with them about whether this is a problem or not. Often in ALF people are very outspoken. They become used to it and are very tolerant. Bickers often are not held onto. My brother likened it to being in a 60s commune and requiring community meetings to work out certain disputes. But he was so tolerant of things that he heard and saw. And of course they tolerated HIM! Bickers over such silly things as when the shades could be up or down in the common room, when the communal room TV had to go off, etc. were generally easily settled and few grudges held.

With dementia loosing "filters" that we all have to use to live a "respectable life" get lost.
This is to be expected and you probably can not change it and it will get worse.
It might be time to consider moving mom from Independent or Assisted Living to Memory Care.
the concern with either Independent or Assisted Living is there is the possibility that a resident could leave the building without much of a problem

I just yesterday received a call from my LO’S AL that they are planning to speak to their staff psychiatrist, because LO has gotten into the habit of grabbing “butts” as they go past her.

LO has survived TWO separate attacks of Covid 19 and an attack of an additional, often fatal bacteria, been moved from her “apartment” on a different floor of the AL, and not seen me, her ONLY family contact, in 6 months.

I’m OVERJOYED that she’s still attempting to assert herself, and of course was delighted also that she be seen by the psychiatrist who has been a compassionate source of assistance in the past.

Your exuberant mom HAS DEMENTIA, but she has survived in a Covid epidemic, is STILL able to leave her residence, and still has the power to express an opinion about what she’s able to observe! HOW FABULOUS!

If you are dealing with professionals OR caregivers who don’t UNDERSTAND HER, please don’t attempt to change HER before considering whether something AROUND her ought to be changing too.

In her earlier years, my amazing LO was a BANK PRESIDENT, NOT a BUTT GRABBER. I AM SO GRATEFUL. That she WANTS to get the attention of those around her. I’m hoping for a REAL VISIT with her in a couple weeks, and if she makes a move on my butt, I’m sure that with the help of her caregivers and her psychiatric support person, we’ll work that out. But I’ll also be whispering a “Thank You” to her Guardian Angel for keeping her around long enough for her to be able to pinch me.

It is normal to keep wanting to interact with our LO as if they are their former selves, but they no longer are. You trying to do talk therapy with her only really helps you (maybe it's not really helping you). She can no longer help her behaviors and has lost her capacity to learn or even stay the same.

Teepa Snow is a dementia expert who has many good videos on YouTube. She outlines how to interact with LOs who have cognitive impairment and decline. Might be worth a look for you. May you receive peace in your heart.

You can sit down and have a heart to heart talk and she will probably agree to be more careful in what she says (she might even be horrified to hear it) or she may become angry and deny it all, but the bottom line is she wouldn't remember any of it nor have the capacity to change even if she wants to. Call it out when you see it as you did at the doctor's appointment but let staff deal with any problems there.