Logistic Regression - Odd Ratio

#1
Hi,

I have the following two logistic regression models where the independent variables are the same for the two models:
logit(Brand1) = b0 + b1 x1 + b2 x2
logit(Brand2) = c0 + c1 x1 + c2 x2

I can't use multinomial logistic regression because the respondents can choose more than one brand (that is, Brand1 and Brand2 can equal 1 at the same time).

My question is, can I compare the odd ratios accross the two models? If, for example I have odd ratio (Brand1,x1) = 3 and odd ratio (Brand2,x1) = 5, can I say odd ratio (Brand2, x1) is higher than odd ratio (Brand1,x1)? That is, can I say Brand2 is more likely than Brand1 to have characteristic x1?

Thanks for your help.