This paper considers the computation of Wyner's common information between outputs of additive Gaussian channels with a common input. The work is motivated by recent generalization of Wyner's common information to continuous random variables and the associated lossy source coding interpretation, as well as its application to statistical inference. It is shown that with independent and identically distributed Gaussian noises, Wyner's common information between channel outputs is precisely the same as the mutual information between the source input and the channel outputs regardless of the source distribution. The result extends the previous result when the source distribution is Gaussian. Generalization to additive channels with correlated noises and its application to statistical estimation are also presented.