A person or entity making a copyright claim need not respond to a request or "demand" that s/he establish ownership of the copyright or authority to represent the owner. Such a person could simply file suit. But a legitimate copyright owner would be likely to respond. If a "troll" actually files suit, s/he would need to establish ownership as part of the suit, and might be subject to sanction for filing a frivolous suit if s/he could not do so.
What follows is largely US-based.
Notice that, in the US, before filing suit, the copyright must be registered, and the records of such registrations are public. This is provided under 17 USC 411
...no civil action for infringement of the copyright in any United States work shall be instituted until preregistration or registration of the copyright claim has been made in accordance with this title. In any case, however, where the deposit, application, and fee required for registration have been delivered to the Copyright Office in proper form and registration has been refused, the applicant is entitled to institute a civil action for infringement if notice thereof, with a copy of the complaint, is served on the Register of Copyrights
One can have the US copyright office search for registrations of a particular work or by a particular owner. (There is a fee for such searches.) You could also do a google image search to find if the image has been posted elsewhere, and who claimed the copyright, if anyone.
Rules will be different if the copyright is claimed to be non-US.
See this question for more on this issue.
17 USC 501(b) provides that:
(b) The legal or beneficial owner of an exclusive right under a copyright is entitled, subject to the requirements of section 411, to institute an action for any infringement of that particular right committed while he or she is the owner of it. The court may require such owner to serve written notice of the action with a copy of the complaint upon any person shown, by the records of the Copyright Office or otherwise, to have or claim an interest in the copyright, and shall require that such notice be served upon any person whose interest is likely to be affected by a decision in the case. The court may require the joinder, and shall permit the intervention, of any person having or claiming an interest in the copyright.
Sections 502-505 (found on the same web page) indicate the various remedies that a court may order.
Section 506(3) provides that:
(e) False Representation.—Any person who knowingly makes a false representation of a material fact in the application for copyright registration provided for by section 409, or in any written statement filed in connection with the application, shall be fined not more than $2,500.
That might discourage a troll from actually filing a suit.