I'm attempting to conduct a meta-analysis using (logged) odds ratios, I'm using the Generic Inverse variance method (Review Manager) as some of my studies only report odds ratios and CIs (not raw data). One study reports the results of logistic regression (only), from my (limited) understanding, β is essentially the logged OR of the bi-variate association, but I'm slightly confused as the standard error reported seems very small (especially in a uneven sample with a rare event for the outcome) and so it appears to be disproportionately weighted in the final analysis. Does anyone know whether the SE in the regression output is the same as the SE I need for the meta-analysis? Or am I missing something vital?! Any help greatly appreciated :)
This depends on what the authors exactly did, so I cannot be sure as I don't have the article in question. I would expect that if one reported odds ratios and their standard errors, the standard errors reported are the standard errors for the odds ratios not the standard errors for the log(odds ratios) (what you called $\beta$).
If you want your $\beta$ you just take the natural logarithm of the odds ratio.
If you want the standard error of the beta you divide the standard error of the odds ratio by the odds ratio.