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Simply put, character artwork is a privilege, not a right, even in an art-centric group. You do not have the inherent right to have your character drawn by someone, and no artist is under any obligation to draw you for free. Ever. So, the best advice when asking for free artwork is "Don't". Good artwork takes talent and time; not just the time taken to draw your picture, but the time taken to draw every picture the artist has ever drawn, which shaped their style and skills for your picture. That's no small accomplishment, regardless of the artist's skill level. So, saying "I want free art!" means "I think art is worthless and artists' skills aren't worth compensating", whether you intended it that way or not.
That said, money's tight for everyone, especially young people, so not everyone has the extra funds to pay an artist a fair fee for a commission. Luckily, there are lots of artists who offer free work! Their reasons vary; usually they're fairly new and looking for subject matter to build their skills, and/or they haven't worked up the self esteem necessary to properly value their own talent (a common problem among artists). Whatever the reason, it's pretty common for artists to offer free work, but never, ever treat it as a common occurance, or take this for granted. Here are some basics when responding to free art offers.
1. Show gratitude. An artist offering free artwork is like a gourmet chef offering free samples, it's a true treat that should be savoured. Thank the artist for their work, and for making the offer at all, and show your appreciation by promoting them to your friends. Don't do it by saying "Hey, this person's offering free work", but something along the lines of "This artist was generous enough to draw me for free, and they did a great job!". Above all, never ever take a free piece of art for granted.
2a. Follow the rules. Almost every artist who offers free artwork puts conditions on that offer, so make sure to carefully and thoroughly read the entire offer and its conditions. For example, there's almost always a limit to the number of spots, so if the offer is for 5 spots, don't whine when you're #6 and get nothing. Sometimes there'll be a contest, competition, or scavenger hunt involved, ranging from easy to complex. If this is the case, and you don't play the game, you're not entitled to a prize. Other times, the artist will be looking for specific genders, species, colour patterns, or scenes, usually to practice something specific. On forums, there will often be a seniority requirement. And, I've even seen artists offer free art only for those who have no artwork of their character at all. Whatever the case, make sure you follow the conditions of the offer, and if you're not able to do so, find another artist offering free art. Don't beg for free art outside the terms/conditions of what the artist is willing to do.
2b. Related to the above point, if an artist is offering art for members of a particularly forum or website, don't sign up for that website for the sole purpose of getting free art. It's not fair to the other members of that community, and it's not fair to the artist. That said, if you intend to be an active member of the community when you do this, so be it, just make sure you're signing up for more than just the free art.
3. Don't nitpick. While every furry artist generally makes every effort to portray your character the way you want, sometimes mistakes or omissions are made. You're free to politely ask for a correction or revision, preferably during the "pencil sketch" phase, but when no money has been exchanged, the artist is under no obligation to honour your request. When it comes to free artwork, you get what you get, and if you didn't pay for it, niceness and charm are the only way to get revisions done. If you're pushy and demanding, don't be surprised if the artist says "tough cookies". On a related note, if you have an exceptionally complex character, or one with very specific details/markings that are crucial to the character's design, free art might not be the best choice for you.
4. If you can't give them money now, do it later. There are exceptions to this, but in the vast majority of cases, artists who offer free art are doing so for exposure, to get their work out into the fandom, with the hope of increased future business. Promoting their work is important, as mentioned in #1, but if you like what they drew for you, it's a very good idea to get a paid commission from them in the future. At the very least, offer a tip of some kind, whether it's money or something else.
5. Be patient. This goes for any artwork, but it's especially true for free work. Any art takes time to create, and even the people who can crank out amazing work as fast as an old printer need time to work. When it comes to free art, there's the additional caveat that the artist is under no obligation to do it at all. You didn't hire them to draw for you, there was no money exchanged and no business agreement, you're getting something that's a gift, a privilege, and a treat. Therefore, expecting it within any sort of timeframe is unreasonable. Even if an artist gave a specific timeframe, delays happen, although you're welcome to ask if they're still planning to do it if it's been a long time. Just make sure you frame the question as "if", not "when", and be polite and gracious about it.
Note: Art trades aren't freebies, they're a commission paid with barter instead of money, and subject to the same rules of conduct as a paid commission. Any artist who treats trades like freebies isn't worth trading with.