Webberstyle said:
Her point was essentially that boo hoo hoo cops have it real hard, and that regular citizens can't be trusted with weapons, because cops have these great background checks that make them stable and immune to mistakes 
Ask her: So, do you think everyone in the entire world who could possibly pass that great background check has done so? i.e. is everyone who could possibly qualify to be a law enforcement officer gone into law enforcement?
She'll have to either admit 'no' and admit that there are plenty of people out there (like her) who are trustworthy enough to be police officers but went into some other profession but can still be trusted to defend themselves and others, or claim 'yes' and then explain what'll happen when current LEOs retire with no one worthy to replace them.
Or, you can debate with her just what it is about the LEO profession that draws the types of characters that made it necessary to do background checks, psychological profiles, cross-examinations, etc, etc, to keep out punks, bad-asses, bullies, etc. Why do cops want to rule over people, and why should they be the only ones capable of enforcing their will at gunpoint?
Then ask her why every jurisdiction has an internal affairs office, if cops are so stable and immune to mistakes & corruption?
About the CCW, tell her 37 states have passed shall-issue statutes, meaning anyone who wants one can carry concealed. Except in Alaska and Vermont, where you don't even need a permit - anyone can carry concealed. In which state are the streets running with blood from all the shootouts between people who aren't as "worthy" as LEOs?
Or, do a google search on case law resulting from LEOs failing to protect, or deliberately not protecting a civilian, and how the civilian lost the suit, BECAUSE LEOS HAVE NO LEGAL OBLIGATION TO PROTECT ANYONE. It's still up to the individual to protect him/herself, and the cops just investigate after it's all done.