Generally one looks at a designer's portfolio to get a sense of their style and abilities.
Offering to pay for a single page is a reasonable thing. Although for a 20p piece that may not really be a huge indicator. In such a piece there's typically a cover design then internal pages to coincide with that design, but internal pages are generally different on the whole. Still it may work. Realize that for a 20pg piece the entire project needs to be considered even for merely a cover design. One can't design a cover with element that won't necessarily work for interior pages. And often, as one works through a 20pg piece, there may be cover changes needed to coincide with the interior design. Generally, a designer considers the "whole" not each page individually. So even if you get a cover design you love, you should be aware there may be changes once the full project is developed.
I know this isn't what you're asking, but it's important: Creative iterations for something like a 20pg piece are problematic. Consider one design may take 20-30 hours of work... so 2 designs.. add another 70% of the time necessary, minimum. Each iteration will create a considerable amount of additional work. That's not something most designers will do free of charge. I know I'd be happy to provide multiple iterations if each iteration was paid for. But, as a client, you are then spending money for things you may not need or use.
A better course of action is to possibly pay for some iterations of the cover design, until you find a cover you are happy with and then the designer can move on to interior content. This way you have the opportunity to nail down a general design you like without wasting so much of the designer's time (and your money).
Honestly, if I were approached with this, I'd direct you to my portfolio and try to communicate the amount of work you are asking for with iterations. It's exceptionally unreasonable unless your pockets are deep. If you don't intend to pay for creative iterations of a 20pg piece, and want them all under that initial estimate/quote of "$xx"... I'd politely tell you to go pound sand.
:) But I'd have no problem with cover iterations until you are happy, again as long as they were paid for.
I would also point out the (general) Paul Rand theory of design... I'm completely paraphrasing...
Design is problem solving. The designer solves the problem of how to get the audience engaged, interested, and excited about the content. A good designer seeks to do this for everything they do.. they want to solve this problem. To that end, there is generally only one "best" solution. Any other solution is secondary and may be lacking in some particular area.
By demanding iterations, here's what actually happens...
- The designer creates the initial design which they feel is the best course to solving your problem.
- A second iteration causes the designer to not make the decisions they think will work best... but rather use secondary choices (as good as I can without doing what I actually want to do). This may be the best attempt to solve your problem for a different audience - that may or may not work if the audience hasn't changed.
- A third iteration will merely be the designer trying to make you happy, not necessarily the audience. Most designers will try and strike some sort of balance between the two, but really, they lean more towards your happiness than effective audience targeting.
- Any further iteration will merely be an attempt to make you happy and the designer has most likely divested in the project and merely wants it to be done at this point.
I know clients tend to feel they get more "bang for their buck" if they can see a few different design iterations. The reality is though, there's generally not a great deal of value in multiple iterations for anything other than branding and/or packaging. (Branding/packaging because, on the whole, they are concerned more with a company's "personality" as opposed to merely attracting a target audience.)
My point being, if the design brief was thorough and communication is good with the designer, you'll get the designer's best work the first iteration. More iterations don't always make for "better" work, unless there's been a lack of communication.
To use @Wolff's meal analogy from comments... Imagine being given 5 ingredients and being asked to cook a meal. You must use all 5 ingredients. That first meal will be good. Then you're asked to make a different meal with the same 5 ingredients.. the second meal may still be okay, but chances are it won't be quite as good as the first. By the time you get to the 4th or 5th meal with the same ingredients, the meals are generally "passable", but rarely noteworthy. Design really isn't much different. There is a rate of decay with iterations.
Many designers will "suck it up" and create iterations when they are asked. But generally only for more simple, one-off, items - business cards, fliers, handbills, etc. For small items a designer may or may not charge for such iterations. Less experienced designers may even offer iterations as a selling point to clients. (Note that doesn't change the bullets above - they are still all true.) More experienced designers typically will charge for iterations. Iterations for multi-page pieces are just unworkable unless the client is essentially willing to pay the full price for each and every iteration. No experienced designer is going to provide iterations for a multi-page piece without some heavy costs involved.
Note that by "iteration" I am referring to an overall design direction change, not merely color alterations, copy corrections, or minor changes such as that.