If a work is "Copyright © year Author One and Author Two" (with no written or verbal agreement between Author One and Author Two regarding regarding ownership), what rights does each individual author have? Can Author One republish the work or create derivative works without the permission of Author Two, or does Author One require permission from Author Two to do this? Any other legal ramifications?
A copyright notice like that could mean the authors are claiming copyright in different portions of the work, or claiming a joint copyright. With the former, standard copyright law applies with respect to each part, so I'm going to examine the joint copyright aspect.
Broadly speaking, joint works tend to be an unclear and internationally inconsistent area of copyright law and you haven't specified jurisdiction. Since this is an English language site and the US and UK are polar opposites on most of these issues, those will be the jurisdictions I examine.
- Definition: To qualify as a joint work, each author's individual contribution must be inseparable or interdependent, and the authors must intend to be joint authors (17 U.S.C. s 101, Childress v. Taylor, Erickson v. Trinity Theatre, Inc.).
- Rights of use: Joint authors can independently exploit and license a work without consent of other co-authors, but have a duty to account profits to co-authors (House Report No. 94-1476 (1976), Goodman v. Lee, Community for Creative Non-Violence v. Reid, etc.).
- Definition: To qualify as a joint work, each author's individual contribution must not be distinct, they must work towards a common design, but no specific intention of joint authorship is required (Copyright, Designs and Patents Act 1988, Art. 10(1), an 11-factor test was articulated in Martin v. Kogan para. 53 by the E&W Court of Appeal).
- Rights of use: Joint authors must seek consent of the co-authors in order to exploit and licence a work (Cescinsky v. George Routledge & Sons, Ltd., Hodgens v. Beckingham).
For further information, see this excellent paper by Elena Cooper comparing how US and UK law diverged on this point.
If there was intent to have joint ownership of the work going in then there is a single copyrighted work and both owners must agree on any dealings with it. Neither owner may use their contributions without such agreement.
If instead there was no such intention (which having "no written or verbal agreement" seems to indicate) then each author retains their copyright to the material they contributed. Each may deal with their contribution however they wish but they each must have permission to use the other author's work if they wish to deal with the work as a whole.
Some works do not lend themselves to collaborative work - two authors working on a novel or research paper may find it hard to disentangle just who contributed what to the final product. If author A reproduces work that author B claims was theirs then there are the makings of a dispute. A joint work overcomes that problem as it is known up front that all author's permission is required for everything.
Others work perfectly fine as a collaborative work - anthologies of poems or short stories, different posts on a single blog, even large sculptures or photo collages.
However, the presumption is that a work is collaborative unless the authors are clear that it is joint.