You know that you don't have to reply on Twitter? When you reply you show that you are willing to be a target. Twitter is not where legal documents are served. In truth, what has happened is that two people who know no better have gotten involved in something they know nothing about.
There's an example of a counter claim document below but you probably don't need it here. Just don't get involved with this kind of silliness.
Here's the thing, bearing in mind that I am not a lawyer, just a webmaster with a lot of original and licensed content.
1) When a person uses another entity's content without permission on a website there are specific steps to follow. In the US you have a set of laws that cover this situation called the DMCA (Digital Millennium Copyright Act). Learn more here: http://www.dmca.com/FAQ/What-is-a-DMCA-Takedown
2) When you post to Pinterest the first point of contact for an infringement is the website and/or host for the content. Your name is not Mrs Pinterest and Pinterest is not your site. This does NOT mean that ultimate responsibility for infringement does not lie with the poster of the image but that the correct route to rectify the infringement is clear and it is through Pinterest. As a user of images on Pinterest you actually have some legal defences anyway. There's not a lot of people being sued by IP owners for stuff posted on Pinterest!
If you believe that the infringement notification passed through Pinterest was incorrect then this is where you can use the counter claim letter. If a counter claim is not responded to within 7 working days then the original infringement notification is voided and, hypothetically, Pinterest should reactivate the image. As a matter of interest, the reason that I started using counter claims was to avoid silliness with my host of the time (Hostgator) who were fair and sensible in responding to complaints. By counter claiming and having the original claims voided I was proving to Hostgator that I was actually a responsible 'citizen' and my account was never affected by the claims I received. I do not know how Pinterest would react however it seems to me that the images you are posting are NOT yours and thus there is, absent procedural irregularities, much of a leg for you to stand on.
3) Notice of an infringement is sent to the host/webmaster in a specific form that should demonstrate that the complainant has legitimate rights to the content being claimed as infringing. In most cases that I have seen that format has not been followed and that renders the complaint invalid. Moaning at you on Twitter is NOT a valid notification of infringement.
4) If you run an online business that uses other people's content then it is a matter of business survival to be aware of what copyright is, how it works, and how you manage it within your business. Right now your business is suffering because of that lack of knowledge and, I am certain, your concerns are infecting others!
In your case there is no action to take unless and until Pinterest remove content from your account AND give you a warning. Pinterest knows the law and how to manage it within their business and to your benefit. If you continue to receive complaints then your account might get closed. That's normal, it is what I do as a webmaster. However, what Pinterest will almost certainly NOT do is investigate claims before acting. They will simply delete the image from your account and leave all other versions intact until a claim is made against those versions.
As a business using a spamming tool and yes, SM is a spamming tool - albeit a low level and relatively innocuous - one should understand that this is what they have and the effect is that one absolutely does not invest emotionally in what are disposable accounts that can be replicated in moments. One of the choices that you made when choosing to use a spamming tool was that much of what happens is automated and you absolve yourself from control.
That's a business decision and you must uninvolve yourself from emotional attachments to your 'babies' the boards that SM creates automatically on your behalf.
On the topic of copyrights though, if I were Jon and his posse I'd be looking to use an image source such as Pixabay: https://pixabay.com/ due to their licensing policy.
To conclude, there is no action for you to take in this regard apart from arming yourself with knowledge and understanding your relationship to the tool you use and the output it creates. It'll help you to take a more constructive attitude toward your use of Pinterest and the manner in which you deal with threats online. There is no action that you need to take unless and until you receive a 'proper' legal document from a proper legal source. Jon's advice about using domain privacy is useful here.
If you, or your host receive DMCA takedown notices in respect of content on your own site(s) then, just as with Pinterest, it is usually easier to delete the image and use another one. On my site's content pages I ONLY use images to which I have explicit licencing paid for by me - I don't use 'free' or any form of claimed copyright free or Creative Commons. Licensing images is very inexpensive these days.
If you receive a letter from another entity about content on your own sites and you choose to contest the complaint then here's a form letter that you can use: https://www.cs.cmu.edu/~dst/Terrorism/form-letter.html Before using it make sure that you understand the basis on which the counterclaim s being made and delete all the irrelevant ones from the document. Understand also that the next step is that the complainant lawyers up and at that point things can start to get costly for you so use this ONLY if you are sure of the provenance of the materials being complained about (hence why I only use fully licensed content on my sites!)
For reference here is the Pinterest copyright infringement page with the form they invite copyright owners to use to initiate a DMCA claim: https://about.pinterest.com/en/copyright
Here's a page specifically about why it is an overreaction to delete boards over concerns about copyright: http://www.lextechnologiae.com/2012/03/03/why-deleting-your-pinterest-boards-over-copyright-concerns-is-an-overreaction/