As Bugsy and Byron have said, consider them a consumable. Dr Slick are not the cheapest scissors available, but considering they have come from manufacturer, to wholesaler, to retailer, to you, and half way around the world in the process they can't be much above them. A really top pair of hairdressing scissors can cost 2000 UK pounds+ and are worth sharpening. My mother was a seamstress, she rarely bought new scissors, when she did they cost over a weeks wages. 4 Pairs lasted her entire working life. Good scissors always have cost a lot.

At the other end of the scissor market, where most fly tying scissors fit, the steal will, at best, be case hardened. This will give a hardened surface to a depth of less than 1/1000". How much material is going to be removed in sharpening them? Probably all the hardened surface and more. Leaving you trying to cut with soft metal. They may be better when sharpened, but will not last.

Tying commercially I find that I am best getting the cheapest scissors I can and changing them quite often. The ones I prefer (Bexfield Embroidery with plastic handles) I can get wholesale for just over a pound each. They last me, at best, a year, at worst, six months. A quick inventory shows me I have around 18 pairs in stock. However they are not serrated. (Well they are not when I get them. Watch this space ).

Sorry to be the barer of bad news but you now have a pair of utility scissors, for tying the best thing is to buy new ones.

Cheers,
A.