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Tribbers generally receive a free trib copy of the zine they trib to. Trib copies may be dependent on the size of one's contribution, though -- for instance, a zine editor might declare that tribbers needed to send a minimum of three pages of story, or a certain number of poems or pieces of art, in order to receive a trib copy.
The tacit, or not so tacit, agreement that a contributing fan to a zine receives a free copy is displayed in this personal statement:
I contributed to [zine name redacted] two years ago. My contribution was accepted... but I never heard any reply from them other than the initial acceptance... I saw this zine on sale at MoreEastly Con, complete with my contribution, so I wrote the editors asked for my contributor's copy. They never replied... I highly object to this zine being sold with my work in it when the editors would deny me my contributors. 
Fans generally agreed that stories printed in a zine needed to "time out" before they could be posted online. This "time out" was usually a year.
From a fan, Lorraine Anderson, in 1995:
You should also know that I have been a ‘zine editor in the past, which is why I found Mysti’s editorial extremely interesting. It recounts an incident she had with one of her writers: apparently this writer put his story out on the World Wide Web some time before it was to appear in Green Eggs and Ham #6. Mysti was quite justified in being upset... this is almost tantamount to prostituting yourself, then representing yourself to your fiancé as a virgin. (OK, my language is a bit strong.) This sounded like a case of ignorance rather than intent, simply because the author did inform Mysti what he had done... but it does make one wonder what less principled authors might do...? 
Susan M. Garrett wrote:
Every writer invented fan fiction on their own. Or at least, in a world where you needed to know someone who knew about fandom or had been to a convention before you realized that people were writing fanfiction, the average fan usually committed fanfic prior to knowing that fanfic, as a concept, existed and had existed for quite some time.
After writing the fanfic, the next impulse was to share it with others. That was easy enough to do – simply loan a copy of the story to your friend. And wait. And wait. And wait. And wait. Until you hesitantly asked them what they thought of it and they admitted sheepishly that they threw it out with last semester's algebra notes. Thus, the idea of actually making copies of your work. But copies would cost money. And even if you did make copies and stapled it, you'd have to sell it to someone to make back that money. But who could you sell to? And why would they want to buy it? The editor became the means by which your work would be read by hundreds and perhaps thousands of people. The editor created the copies, paid for the copies, and sold the copies, taking on the financial risk and, possibly, the financial reward. And in addition to the joy of knowing that people were reading your fiction, you got a bonus . . . a free copy of the fanzine in which your work appeared. Not all fanzine publishers gave contrib copies to all contributors. Sometimes the contributor received a discount on the purchase price of the fanzine. Sometimes the size and form of the contribution would determine whether the contributor received a gratis copy or a discount (a poet might need more than one poems or filk printed in a fanzine to merit a free copy). Sometimes a contributor might receive a number of free copies in exchange for their submission, usually for a large contribution (like an entire novel) or a major piece of artwork (like a cover). But the contrib copy always served as a reward for having contributed work to a fanzine, as well as an incentive to contribute in the future.Contrib copies might be mailed to a fanzine contributor or might be delivered directly to the contributor at a convention of through an intermediary. When the average fan could spend hundred of dollars a year on purchasing fanzines, being able to earn that copy through a contribution of your own work was an idea that appealed to many fan fiction writers and fan artists. 
Some Fan Comments
Let's talk about that much-vaunted perk, the contributor's copy. "You're getting a free copy of the zine!" some cry, in the same breathless tone they'd use if we'd just won the state lottery. But that copy is "free" in about the same sense that freeways--at a construction cost of millions per mile--are "free." This belief that the artist is getting something for nothing, or damned near it, fails to take into account the costs involved. Cruised an art supply store recently? Then you know how expensive the materials are. References--you know, photos and such-can also be expensive; although sometimes editors or friends can provide 'em, they may not be appropriate to the story or scene (have you ever tried to illo a tense, dramatic scene using a stock, smiling-straight-into-the-camera publicity still of Our Hero? It's not a pretty sight....). 'Stats or halftones are often necessary— usually the artist picks up the tab. And of course there's the postage to get the art to the editor. I recently did one illo for a fanzine. Photo references were $20.00, materials about $2.00. Postage to ship the art to the editor was $4.50. I'm out of pocket $26.50 for this "free" fanzine even before I take into account the time involved in working out and completing the illo, and packing it off to the editor. For another zine, I did three illos. References were only about $15.00, as I already had some of them to hand. But the halftone and 'stats cost me $25.00, postage another $5.00 or so, bringing my costs for that zine to at least $45.00. Again, this is before figuring up the hours involved. And there are many of those, since doing an illo usually takes considerable time and patience (unless you're Suzan Lovett, who is able to do an elaborate drawing, start to finish, in the time it takes the rest of us to unpack our pencils)! 
There seems to be a lot of confusion about contributor's copies, mainly, who gets 'em! I think there are a lot of different policies out there, and sometimes they even vary from issue to issue of the same zine (depending on the size and expense of the zinel). Ideally, all contributors received a free copy of the zine, postage paid. The key word is "ideally"... When the zine assumes heroic proportions, and the costs mount wildly, somehow it doesn't always seem possible to send a free issue to, for example, a fan who contributed one poem -- and lives in Denmark! (Overseas postage is a killer, usually costing more than the value of the zine; I love it when overseas contributors offer to reimburse me for postage!!) I think to avoid misunderstanding (which seems to be the reason this question kept coming up), contributors should make it a point and ask what the policy of the zine is. Zine-eds don't aIways remember to tell every contributor, so ask! With the WC, we try to give a free copy to every contributor -- but sometimes (the old "one poem" case), we just can't justify it. Then we offer a half-price or something similar. And, as I said, we really appreciate the understanding of our overseas fans. After all, if we go bankrupt, there won't be a zine anymore! 
- from Datazine #38
- comment in The Hologram #9
- from The Fantastically Fundamentally Functional Guide to Fandom, specifically "The Fantastically Fundamentally Function Guide to Fandom for Fanzine Readers and Contributors" (1989)
- from the editor in her zine, Psst... Hey Kid, Wanna Buy a Fanzine? #2
- from Southern Enclave #16