you could ask "wer ist zuständig" if you want to know who will clean up the mess. if you come to the scene and only see fragments you can ask "wer ist verantwortlich" but in order to not out yourself as a foreigner the good ol german "wer ist schuld" is the better choice, because we are obsessed with blaming someone.
"zuständig" cannot be used to describe someone's character. "Anständig" would be the word you are looking for. "zuständig" can be used in a legal/administrative way such as e.g. "the department responsible for tax refund" - "Die Abteilung zuständig für Steuerrückerstattung". As the sentence appeared out of context I would translate "Ich bin zuständig" as "I am in charge".
The way I see the difference between these two sentences is: "Wir sind verantwortlich" = "We are responsible (for doing it)" "Wir sind zustaendig" = "(You can trust us), we are responsible (people)"
Anything that you see wrong? Please note that this is as much for me as anyone else!
"verantwortlich" is more individual / personal and not bureaucratic. A person can be "verantwortungsbewusst", someone who thinks before he acts, who is aware of the consequences of his actions. A person can act "verantwortungsvoll".
"zuständig" is rather an official or professional and not a personal matter. It has more the meaning as in "The offices are arguing over jurisdiction." As you may know we have a complex social system in germany and in some cases it is not quite clear who is "zuständig". So the offices argue who has the "Zuständigkeit" - who is in charge of the case.