Because my licensure area (school media centers) is a special area in which students are not graded, formal assessments and scoring will seldom be required. An individual teacher may require students to show proficiency in research or information literacy tasks, in which case the librarian should certainly consult on the best methods and practices for assessment.
In elementary grades, many librarian-driven lesson plans focus on basic activities such as finding or locating a book. This may involve searching the online catalog (technology skills; thinking skills, or how to conduct a search) as well as physically locating the book on the shelves (fiction versus non-fiction, alpha by author's name, Dewey Decimal system).
Performance assessments where a student is asked to perform a task or conduct a search work well in this environment. Informally, the librarian can check to see if students are progressing at these basic tasks. The librarian watches a student as he/she conducts a search of the online catalog, helping point to the correct links or shaping more effective search phrases.
When a student asks the librarian for help in finding a book, she can guide the student with a series of questions that allow the student to generate more of the solutions or steps. The librarian can then understand where the "disconnect" is for each student based on the feedback she receives. The goal of such informal assessments or "check-ins" is to see that the student is able to perform searches or finds books independently. Students will know if they are learning how to search or find books if their efforts produce the book or material they want.
In more formal assessments, the librarian would check the students' knowledge about the library and how it works. One fun way to test this, particularly in younger grades, is with the use of a scavenger hunt. The Association for Library Service to Children (ALSC) provides several examples of grade-appropriate scavenger hunts for use in your library. (My favorite question for grades 2-3 is "What is the name of the children's librarian?" Always an important question! And it addresses the need for a librarian to have great relationships with all patrons!) I will note, however, that in older grades or in college reference situations, scavenger hunts - especially ones where the teacher is simply giving the students a busy-work activity to complete - are not optimum. And they are often hated by librarians. Note to teachers: ALWAYS consult with your librarians on library and research assignments!
As students grow, basic library skills are a must, and the teachers can then assess students on more complex research and information literacy tasks. Extended research projects, annotated bibliography assignments, and the like can show if a student understands not only that a need for information exists, but what type of information is needed, and how that information can be read, understood, and synthesized to satisfy his/her needs.
For more information on how libraries affect academic achievement in all grades, I leave you with this quote from the American Library Association's Add It Up advocacy and awareness campaign:
Students whose library media specialists played [an instructional] role—either by identifying materials to be used with teacher-planned instructional units or by collaborating with teachers in planning instructional units—tend to achieve higher average test scores.(from Lance, Keith Curry, Lynda Welborn, and Christine Hamilton-Pennell. 1993. The impact of school library media centers on academic achievement. Castle Rock, CO: Hi Willow Research and Pub.)
The American Association of School Librarians (AASL) also provides this bibliography on the topic of information literacy in school libraries, illustrating the push to develop more standardized assessment tools.