HMMMMM...what is the best thing to do

 To be perfectly honest I don't think you've taught her honesty, I think you've taught her a "trade" command.  She knows if she brings you something she suspects is not hers/she's not allowed to have, you will take it away and give her something else.  Not giving her something when she took the bread out of the rubbish is one example of intermittent reinforcement - she'll keep trying this in case you'll give her something. 

My dog frequently brings me things he suspects, or even knows, are not his.  I don't believe that's honesty, it's because through repetition he has come to realise that sometimes if he brings me a remote control/chinese food menu/sock that I'll trade it for something better.  He's a lab, so "something better" really means food.

She's bringing you Duffy's treats not because she wants to be honest about it, but because that's what she could find.  She may even have smelled him on it and was bringing it to you to show you what she wants, or to get permission to eat it.  Again, my dog will show me things he's found and if I DON'T take them away from him, he assumes that he's allowed to have it.  You've now (inadvertently, although it's not a bad thing) taught her that bringing you things might get her something.

To answer your question, only give rewards for bringing you treats or anything else she shouldn't have intermittently.  Maybe once every 5 or 10 times.  She'll still do it, just in case this time is the time you're feeling generous.

Just my opinion. :) 

GoldenAC, you and I may be in the minority here, but I agree with you that dogs know the difference between truth and a lie.  Altho I'd break it down to the word "trust" (Selli trusts that you will reward him for bringing you something he knows isn't his own -- subtle difference but he knows you see reality clearly and his efforts aren't in vain). 

My example is food -- when I give a dog medicine (and I tend to rescue/work with sick dogs a lot) I NEVER "lie" to them and try to slip a pill in something to 'hide' it.  first off, I think it's insane to try -- because dogs' noses are SO beyond ours that OF COURSE they **know** something has a foreign object in it -- they can SMELL it -- but if the item is yummy enough they will simply choose not to 'object' to the foreign object -- but don't for an instant think they don't 'know'.  That's ludicrous.

So, I always chose to work with reality -- I show them the pill, and I tell them I'm putting it in ... whatever (and that tends to be individual to the dog --from braunschweiger to babyfood to butter and beyond -- whatever works with this dog and this particular med).  But I give them a bit of the 'good stuff' and then SHOW them I'm putting the pill in and they trust me now to let them know when something is 'ok'.

Because sometimes with meds, either it will make them sick to their stomach or be unpleasant somehow *later* and I choose to tie it to the idea "this is helping you".  Training them, literally, to TRUST me and TRUST the vet to make these decisions for them.

 So I'd attack your problem from a trust angle.  Similar to what you've done -- rather than reward a 'chewy' with THAT chewy (and I don't give my dogs rawhides -- sorry *grin*), you could give any number of other treats.  From 30 seconds playing with a toy, to some other high value treat.

Guaranteed tho -- the 'reward' will have to be as high value as the coveted 'chewy' -- because once he decides he's not going to get THAT thing the behavior will diminish.  You will have to occasionally offer the 'chewy' he's brought you AS the reward in order to retain that behavior. 

So, if you want to explain it from logic/trust standpoint -- right now he trusts you to GIVE him the chewy AND he also knows you'll be pleased with him for bringing it to you.  But if he decides you aren't playing 'fair' with him (and justice/fairness are also concepts dogs don't really share b/c what's mine is mine and what's yours is essentially mine too) -- and if he thinks you don't 'value' that thing as much as HE does, he may start getting ahead of you and taking the chewy into his own paws occasionally. 

My bet is that emotional reward may figure high in this.  So the bigger deal you make out of his return of Duffy's chewy and the more PRAISE you connect with it the more successful you will be in the switch.  In other words combine the 'high value' treat you give in exchange with the VERY high value of your being 'pleased' with him -- two rewards for one behavior.  THAT is a good deal.

Benedict

 To be perfectly honest I don't think you've taught her honesty, I think you've taught her a "trade" command.  She knows if she brings you something she suspects is not hers/she's not allowed to have, you will take it away and give her something else.  Not giving her something when she took the bread out of the rubbish is one example of intermittent reinforcement - she'll keep trying this in case you'll give her something. 

But isn't that exactly what honesty is?  She finds something that she knows she should not have and if she wanted to, she could eat these things without anyone knowing, but instead she brings them to me and asks what I can only imagine of as permission to eat them.  The thing that really gets me is that she could just eat these things, which I assume is what she is after, without me knowing, but she seeks me out.

I've found dogs are quite capable of asking, demanding, manipulating, and doing what they darn well please, as well as being purely conditioned. One concept is social, another concept is behaviorism.

IMO, both concepts are present when dogs live with us as family members.