For gods sake please rehome to someone you know personally, take your dog to a no-kill rescue or work with people such as myself to get their help in advertising (thus nationwide) and homechecking for you - as you'll see as you read on, thats what I and many others do and we can work with you to do the best for your dog and for you.
Also, NEVER give away pedigree papers (or the new owner may decide to sell the dog on to someone wholely unsuitable for a big profit), and NEVER rehome unless the dog is neutered/spayed first (or he/she could be bred, year after year, literally to death or until too old and then killed or thrown into the streets).
Trust me, this DOES happen. I see it regularly in my work as an independent network volunteer.
What I do is this - I get calls/email from not just pounds but also owners or those who know of owners who no longer want their dog. Some genuinely can't keep the pooch, some are utter heartless, thoughtless bastards. Sorry to be blunt but I do meet them.
My job is either to find the dog a place in a no-kill rescue, which will assess, train, neuter, chip and vaccinate the dog and then match him to a new owner, who will be homechecked, or to find him another suitable place. A no-kill rescue will keep any dog in their care for life if he can't be homed for any reason.
Otherwise I may find him a foster place with someone who will do the same thing (from a wide network of contacts across the country), or, if he is a family dog with no issues to iron out, possibly a new home directly, which again one of my trusted and reputable contacts will homecheck.
There are quite a few of us who do this. One of our main internet contact places is the website and forum of "Rescue Helpers Unite", which consists of people like me, rescue owners and checked and approved foster carers.
That website has recently taken a new form and I have had little imput into its new one, so "Spangle", my user name, has little to say for herself on there to show you what I do. However, the old site, I believe, still has the old forum accessable so you will see what I do there.
If you visit and register on the new one you can post details of your dog there and we will do our best to give you the help described above. Please do pm me if you'd like to when/if you join the forum, I'm happy to help and give you my email address etc so I can explain further and reassure you that although I must be as mad as cheese doing what I do, I'm largely only a fanatic about dogs and that I am genuine!
You can also register on the forum of my local rescue, for whom I volunteer and look me up there - the internet forum is "Poplar Farm Kennels" a bugger to find by Googling, but please perservere, and I am known as "Harvey" on that site.
If you'd rather contact me another way, please say and I will set up a new email account and give the address here (I say a new account as I am anxious not to mess up my usual one with loads of spam etc). Again, I'll happily help and put you in touch with organisations which will vouch for me.
May I please ask why you need to rehome your dog? I only ask as I may perhaps be able to give advice/put you in touch with others in order to help you keep him if thats what you wish.
Also it would help if you could tell me his breed or what he is crossed with as I may be able to suggest organisations or contacts who can help.
I do hope you'll feel able to come back and reply and can only repeat that you would be VERY unwise and potentially put your dog at huge risk if you don't homecheck and also that it is possible and much easier for you to obtain help in doing so.
Thanks for giving me the chance to make these suggestions.