I have a different answer.
Although your question asks "Who is responsible for 1099," a different (and potentially more important issue has bubbled to the top.
... your need to properly classify the students in their relationship.
IRS actually provide four alternatives:
An independent contractor
An employee (common-law employee)
A statutory employee
A statutory nonemployee
The critical issue still becomes classifying the student. (what sort of reporting?)
After that, who does the reporting, then, becomes the issue.
IRS does provide SOME criteria for making the decision:
Facts that provide evidence of the degree of control and independence fall into three categories:
Behavioral: Does the company control or have the right to control what the worker does and how the worker does his or her job?
Financial: Are the business aspects of the worker’s job controlled by the payer? (these include things like how worker is paid, whether expenses are reimbursed, who provides tools/supplies, etc.)
Type of Relationship: Are there written contracts or employee type benefits (i.e. pension plan, insurance, vacation pay, etc.)? Will the relationship continue and is the work performed a key aspect of the business?
See this: http://www.irs.gov/Businesses/Small-Businesses-&-Self-Employed/Independent-Contractor-(Self-Employed)-or-Employee%3F
What you have here is not unlike employee leasing, where ...
Under Code section 414(n), a leased employee is defined as a person who provides services to a recipient company pursuant to an agreement between the recipient company and your organization, under the primary direction or control of the recipient company
In these situations, the recipient company provides benefits (and W-2's).
The most pertinent two statements here, in terms of you managing your own risk are the following:
(1) "If you go ahead and pay them as independent contractors, and report their earnings on a Form 1099, you, the organization, risk penalties of up to 100% of the withholding that would have otherwise been required to be withheld!"
(2) "If you would like to have the IRS determine whether they are independent contractors or employees, you can complete a Form SS-8 and they will make that determination for you. But based on your fact pattern, they are employees"
Hopefully, we have provided enough information to let you do the necessary research to make an informed decision.
I THIS professional's opinion, you need to ask these questions of IRS.
I would rate the expert here, who has raised the most salient and pertinent issues for you and that is NOT me.
It's the second expert, who has point out some VERY important issues