The positive might be the female Russian spy who was ejected from America for spying. It was positive notoriety for her, because in Russia she got her own TV show.
The negative might be the stigma attached to a bad reputation, like the 18 year old British nanny who was tried and acquitted, of killing the American child in her charge by shaking the child too roughly. Being found not guilty is not the same as never being accused.
Adj. 1. notorious - known widely and usually unfavorably; "a notorious gangster"; "the tenderloin district was notorious for vice"; "the infamous Benedict Arnold";
ill-famed, infamous
disreputable - lacking respectability in character or behavior or appearance
http://www.thefreedictionary.com/notorious
Deno·ta·tion (dn-tshn). http://www.thefreedictionary.com/denotationn.
1. The act of denoting; indication.
2. Something, such as a sign or symbol, that denotes.
3. Something signified or referred to; a particular meaning of a symbol.
4. The most specific or direct meaning of a word, in contrast to its figurative or associated meanings.