Each singular countable noun will need an article unless they are the same part of the sentence (ex. both the subject). So for example:
- A painter, musician, and dancer walk into a bar. (...Ouch!)
In this example, painter is sharing its article ("a") with musician and dancer. But after the verb, bar will need it's own article because it's a separate part of speech.
It makes a difference because without it, an indefinite singular noun could appear to be a proper noun. So, your sentence "Painter is an artist." appears to be about an artist that is appropriately named Painter. :-)
Hope that helps :-)