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Own, as in to own means poseer (yo poseo, tú posees...) although in Peru it is often replaced by the verb to have (yo tengo, tú tienes...) which is a common peruvian misuse of the verb.
Own as in a house of my own, or as in my own house, can always be translated as propio (m) / propia (f). (Gender related declinations are always a problem for those who have English as native language.)
I would like a place of my own, Yo quiero una casa propia.
you have your own car, Tú tienes tu propio carro.
would you like a drink of your own? ¿Quieres un trago propio?
Although in the last case it doesn't sound quite right, it would be best to say:
would like your own drink? ¿Quieres tu propio trago?