A big unknown, that affects how you sell it, is the value.
The larger it is and the closer the color is to the most valuable grade, Fancy Vivid, the more challenging it may be to sell privately, and the more likely you are to need a professional auction house.
Christie's and Sotheby's will take a substantial commission, but they have the trust of, and access to, the most well-heeled buyers.
They would be secure, safe and private for you, and you'd be confident the final price is 'appropriate', as in not too low.
Buyers who wouldn't dream in a million years of buying this from a private party will feel confident buying through Christie's and Sotheby's.
Still with so much money at stake, (I may be crazy) but I couldn't resist first take a shot at selling it myself.
But of course a private party cannot command the high price that a reputable professional can, so it could be argued that private party sale is a very very bad idea.
You posted your HW pink has a GIA report.
That's essential since GIA is the gold standard for grading FCDs where a tiny difference in color can have a huge effect on value.
If the GIA report is over, say, a year or two old I'd get the stone unmounted and returned to GIA for a new report.
If you need to ship it to GIA in NYC or Carlsbad, California I'd use http://www.malca-amit.com
Your job as a seller is to allay very legitimate fears and suspicions and instill confidence in what you are selling.
GIA and Hofer do that for knowledgeable buyers of FCDs.
Next, I'd try to arrange an escrow service to handle the exchange of the stone and the funds to protect you.
Here are some sites where private parties sell diamonds.
I think they are either free or have low fees.
This private sale may take a long time, if it works at all, and will involve risk.
People vary.
You may not have the stomach or patience for a private sale.
It would be totally understandable if a big auction house was your choice.
Well, since you mention the possibility of keeping it- I'd keep the pink diamond and have it made into a beautiful pendant. Pink diamonds are getting harder to get and more expensive. So, if you like the diamond-keep it!! If it harbors bad memories-look at some of the options mentioned above for selling it.