Hitchhiking usually implies that no money changes hands, no monetary transactions for the ride.
How to deal with requests for money beforehand
In Western Europe and the United States it's rare that people ask for money. It can happen though, especially in Germany, where Mitfahrgelegenheit is quite popular. As a rule guaka refuses these rides, though once he accepted a ride because it was later at night after having waited for a long time already and he didn't feel like waiting any longer.
In some countries payment for rides is quite normal, especially for locals. In Romania for example you probably want to ask if the ride is free (how to ask this in Romanian?). In Turkey it's rare, but it happens that drivers want money, just say "yok parra" in such a case.
What to do if a driver requests money unexpectedly
What to do if a driver offers money
Some hitchhikers (such as guaka or quarim) refuse any offers for money from the drivers. Sometimes drivers are insisting, but if you have enough money to go by and you think it's inappropriate to receive money, you could tell them you actually have plenty of work (and income), that you mainly hitchhike for fun, social contact, and ecological reasons. Instead of money driver very often offer you material things such as food and drinks or will give you a CD, flowers, clothes...you can decide if you want to take it or not, in some countries it is unpolite not to take it.
How to hitch with no money
There are some people out there who hitch hike LONG distances with no money such as the infamous Tom Thumb, who hitched from England to India with just his rucksack, 2 loafs of bread and £0.00. There are also not so famous hitchers like Aaronishappy who hitched from Norway to Morocco, with just €2 arriving in Norway and his rucksack (and coming back with 20NKR, 5EUR, 4SF and 2GBP, which is 20NKR, 5EUR, 4SF and 2GBP more than he had when he left!).
While completely crazy it is certainly possible, but not necessarily fun. You will ask yourself several times a day "what am I going to eat", you will, and regret ever leaving your house, but knowing that no matter how bad things get they can only get better is a certain comfort.
How do you do it then? The easiest answer is that you don't think about it, and just do it. The easiest, and probably the only way of survival is to come into contact with people and with hitching it's of course inevitable! After sticking your thumb out and finally getting that lift you've been waiting for, this is it. This is your time to shine and score some food. Most of the time you never need to show to their faces you have no money (but you may look like it) and they will give you something, ranging from clothes to joints to a dinner at their house. The trick is to be as social, open and friendly as possible! They wouldn't have picked you up if they didn't like you after all. If you've really tried to give the best impression you could, then the only choice to resort to is the pity game. Basically tell them you have no money, but not so bluntly. Be "covert" about it and ask questions such as where can you eat for free, use the internet for free, or "Hey do they have homeless shelters here?" Most of the time they'll ask why and then bingo! "Because I have no money, sir/madam" They'll usually be surprised that you really are skint and will be happy to give a hand to the weary poor traveller, usually in the form of money or food. Often in the afternoon, they will invite you back.
However! Every time you hop your jolly arse into that car please don't expect a treat everytime. Because there WILL be many, many times when you get out of the car as hungry and miserable as you were when you got in. The trick is to be genuine in expecting nothing, because when you expect nothing, the smallest thing you get, even if it's a piece of bread or a cigarette, will make you happy.
As for sleeping when you're not invited back, best bring a nice thick sleeping bag and find yourself a good spot to sleep in. In cities, the best spots are industrial zones (although usually a long distance from the centre), upper class neighbourhoods, and of course train stations! Train stations however will often close in the middle of the night to clear the place out of bums like you. Big stations that stay open all night such as Paris and Berlin will often have security walking about to wake you up. If you're stuck in the city centre with no station or access to an industrial zone/posh neighbourhood, your best bet is to keep walking until you reach a quite area and find a nice alleyway or empty park. Just please never sleep bang on in the city centre in clear light. I did ONCE and out of the 100+ times I slept outside, it was the only time I got robbed. If you really have no dignity you can always go to a homeless shelter, they always provide a bed and usually food. They are everywhere in London and Germany, and quite common elsewhere in the first world.
Remember to keep clean! Even when dirt poor you can easily keep clean. Bring a bar of soap and wash in the toilets. People might be a bit weirded out to see some backpacker standing in the toilets in their underwear at the sinks, scrubbing themselves with soap, but hey you gotta do what you gotta do. I never had a problem with it. Anyways, the amount of times you get invited back for a shower is crazy so you shouldn't REALLY worry about it.
If you ever find yourself stuck in a big city, with of course no money and seemingly no way out then your only bet is to talk to people in the street. Talk about travel related things and expand on them (ie directions, where to hitch, etc, use your imagination!). I've asked for directions to the next town and ended up staying at someone's house for the night and leaving with €20! There really shouldn't be an article on how to make friends, it's self explanatory.
If you're hitching in a Muslim country you can basically disregard all of this as Muslim hospitality will clothe and feed you without you ever saying a word!
And the point in doing it? Well, so you can return home after a long adventure without ever having to spend a cent of your money :)