Personally, I never hesitate in straightening coins as I never intend to sell them. My straightening processes run from cold pressing to the full-blown annealing process described below - it all depends on the coin and its condition and careful assessment is needed before proceeding. Anyway, on with the motley...
Disclaimer: the method described below works for me but anyone who takes on the straightening of their coins does so at their own risk and any damage to coin value, self, pets, property or neighbours is entirely their own responsibility.
This little fellah was in a bad state - bent right over and with a nasty split - and annealing was the only answer.
Stage 1
Equipment used: old saucepan upturned in a bowl of water, kitchen blowtorch, wooden spoon, wooden chopping board.
Place the coin on the base of the upturned saucepan, heat it with the blowtorch until dark red and then flick it into the water to quench (some say you should let the coin cool naturally but I've never experienced any problem using the water method). Place the coin on the chopping board and gently ease out a couple of millimetres of the bend with the spoon (or other wooden or non-abrasive implement).
Repeat this process until the coin is flat. I find a wooden spoon ideal for the process, with the round handle good for teasing out gentle bends and to act as a kind of rolling pin to finish it off.
When satisfied, give the coin a final annealing then cool.
This is what the coin looked like after this stage:
Yes, it's flat, but...bit red, innit? Looks awful, in fact, and one of the problems with annealing is that the copper content of the silver rises to the surface. It can be counteracted by "pickling" but this process involves the use of nasty chemicals, none of which are for me, so I try to reduce the effect as descibed below.
Note: sometimes there will be virtually no reddening at all and other times the coin will blush like it's been told a really filthy joke - it depends largely on the purity of the silver.
Stage 2
Materials used: bicarbonate of soda paste
To help remove the red/reddy-grey/grey patination, you can use such as lemon juice or water & foil, but I like the bicarb paste as it's gentle and you have full control and can see what's happening at all times.
Make a thick-ish paste of bicarb and water and rub it gently over the coin until clean (or as clean as you can get it).
After rubbing with bicarb paste:
Still a little pink and of course we've lost the original patina but ,we can bring that back to an extent though it's not likely to match the original.
Stage 3
Materials used: bleach and a pencil rubber
Dilute one part bleach to two parts water and soak the coin until it turns black. Then rub the pencil rubber over it to expose the silver in the high spots. You can use an ink rubber but I find a pencil rubber more effective, albeit slower. If you decide you've overdone it and it's too bright then pop it back in the bleach and repeat the process.
If you don't like the idea of using bleach, try sticking the coin in a hard-boiled egg. It might take a little longer to darken the coin but you have a handy snack afterwards

The final stage is to soak the coin for a few hours in de-ionised water to remove any chemical residue.
Finished result:
It still has a slight tinge of red which looks more pronounced in the image than in reality. Honest.
I'm happy-ish with the result, though there are a few tweaks still to be done on it, but overall it's a vast improvement on what it was.
Footnote: The use of boiling water in coin-straightening is pointless. It simply doesn't raise the temperature anywhere near that required for the annealing of silver and if you've used boiling water "successfully" then you might as well have saved yourself the bother and cold-pressed the coin in the first place. That's not my opinion - it's science.

Hope this has been of help and if anyone has suggestions to improve this method then please do shout and let's see if between us we can get this process nailed

EDIT:
Some great advice in the comments below which are well worth a read. They include:
Annealing should be done in low light to better see when the coin is at the correct temperature
Use various-sized rollers to ease out the bend
Re-patination can be achieved by placing the coin in a box full of rubber bands. Who knew?

Low-quality/debased silver requires special care and attention before proceeding.