Declaring it separately means you can reference it in unit (or integration) tests. If you end up needing to change the contents of the string, you then only have to change it once (where it's defined), rather than making the same change in the other instances of the string.
There can be other benefits to centralization - for example, if you end up needing to add localization (other-language translated versions of the user-facing content), it's significantly easier to hand off a few files with all of the strings to a translator, rather than an ad-hoc bunch of spots throughout the code.
If you're not writing test code, you probably want to look into doing that - it's generally considered a best practice. If it's a quick personal project, it's OK to take the shortcut of hardcoding - but it's definitely not something that's generally recommended.
Edit: On rereading this, I realize that I'd missed a specific piece of your description: that the strings are currently defined publicly in the class. I had had the impression that they were defined externally, in a different file. My recommendation would be to move them out to a separate "Error Messages" class that contains all of them (which is what I'd thought you were describing from the start).