Hello, the right sentence is "spero di essere vivo" for a male, or "spero di essere viva" for a female, because the past participle of the verb vivere changes the ending vowel according to the person. The sentence "speriamo di essere vivi" would be right if you were talking in the plural.

'vivo' in this case is used as an adjective, with English meaning 'alive'. Another example: La torta e` cotta. Here 'cotta' (i.e. 'cooked') is an adjective. Torta: cake.