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We consider both the expectation and finite-sum settings under standard assumptions, and use both classical stochastic and SARAH estimators for approximating function values and Jacobians. In the expectation case, we establish $\\mathcal{O}(\\varepsilon^{-2})$ iteration-complexity to achieve a stationary point in expectation and estimate the total number of stochastic oracle calls for both function value and its Jacobian, where $\\varepsilon$ is a desired accuracy. In the finite sum case, we also estimate $\\mathcal{O}(\\varepsilon^{-2})$ iteration-complexity and the total oracle calls with high probability. To our best knowledge, this is the first time such global stochastic oracle complexity is established for stochastic Gauss-Newton methods. 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