I think these are words/expressions that may come closest to what you ask, but with somewhat less emphasis on the "self" aspect:
to be a fearmongerer
to be a scaremongerer
to be a worrymongerer
The word "monger", meaning to spread, sell, peddle, or push something on someone else, can be used. The word can be used on its own or in combination with other nouns as an ending (for example - worrymongerer, worry-mongerer, or worry mongerer). It usually implies that the person already has the attribute being spread (fear, scared, or worry).
Also, it occurs to me that the word "alarmist" also might work as a person who spreads alarm.