On DreamHost, the “equivalent” of your “public_html” directory is whatever directory you indicated, when you added the domain in the web panel, in the “Specify your web directory:” field. It is in the form of “/home/username/something”, and by default (unless you change it) the something will be “yourdomain.tld”.
This is the directory into which you place your files that you wish to have accessible from the web (as you would in “public_html” on your previous host). Just like can your previous host, you can also put file in subdirectories of that directory.
You can confirm the currently used directory, or change it, for each of your domain/subdomains by going in the web panel to Domains -> Manage Domains, and clicking the “Edit” lilnk under the “Web Hosting” column for the domain or subdomain in question.
You might want to re-consider that plan, as you will likely incur considerable penalties with Google for duplicate content. Even if your purpose in doing that is “legitimate” (you want both .com and .org to produce the same content so a user doesn’t get confused if they type the wrong url, etc.), Google will see you serving duplicate content from two domains as a “SEOTard” kind of "search engine gaming.
The correct way to do this is to use a redirect to “automagically” send a user who types in “sudomain1.yourdomain.tld” to “subdomain2.yourdomain.tld”. You can do this via an .htaccess file, or from the Mange Domains section of the web panel.
If you really do want to have the same content served from two different subdomains using the same directory, there are ways to do that too, but I don’t advise it and am not particularly keen on helping you do it (there is too much of that on the web already!)
Why do you want to serve identical content from two different domains?