I think this is an interesting question and it has brought out some interesting points.
In general, I think all inclusive or all exclusive statements and rules are not a substitute for good judgment and brain power.
If your home has been hit in the past.... dog or no dog.... it's certainly reasonable to be concerned about it being hit again. Though it's certainly reasonable to think that a good dog with alert and protective instincts would be of great benefit should someone come around again, I'm of the school of thought that relying on the dog to do more for you than alert you to an intruders presence is asking more than you should. (i.e. it's my job to defend and protect my dog, not visa versa). So I'd say your instinct that your dog is no match for an intruder armed with a firearm is spot on.
I also commend your concern for the safety of your dog sitter friend. And though people often debate whether or not they are their brothers (or sisters) keeper, the fact that this gal is helping you out with a friendly service, regardless of whether or not you pay her.... (I'd suggest a generous gift certificate) puts at least some responsibility for her well being in your court.
Some thoughts I would add....
1. I suggest that it is better to share with someone your concerns and ask about their preferences, rather than making decisions for them.
2. I would never consider arming someone unless I knew they were competent to use the tool. Since you had been to the range with her, you had a reasonable basis to make such a judgment.
3. Leaving a firearm out leaves open a greater possibility that it could fall into the wrong hands, should your sitter be out of the house for a while (run to the corner store, walking the dog?)
4. Based on all of the consideration above, an how the conversation went with the gal (yours obviously went very well) I might offer to leave the shot gun in the bedroom closet for her as that should certainly be adequate for her to defend herself within your home.
without making a big deal about it, I do thin that this comment....
Probably in ever state it would be illegal for her to be in possession of a handgun not registered to her.
does offer some insight into how ones mindset can change when you get used to a certain amount of obtrusive government and the gradual loss of liberty. Because the day after a law is passed we continue to go to work and attend to our families and business, we can be easily lulled into thinking that it's "no big deal" and that the new normal we've become accustomed to, must certainly be normal for folks living in other parts of the country as well.
And finally... regarding liability... looking out for someone else will usually incur some form of liability. But then again, turning your back on a friend who's helping you out isn't entirely safe either. Just how thrilled would you be to go to bed every night in your fine home, if perhaps, your friend was brutally raped and murdered in your own bedroom, just a week prior. A steam vac may work wonders on the carpet. Perhaps not so well on your conscience/soul.