We present the properties of X-ray detected dust obscured galaxies ( DOGs ) in the Chandra Deep Field South . In recent years , it has been proposed that a significant percentage of the elusive Compton-thick ( CT ) active galactic nuclei ( AGN ) could be hidden among DOGs . This type of galaxy is characterized by a very high infrared ( IR ) to optical flux ratio ( f _ { 24 \mu m } / f _ { R } > 1000 ) , which in the case of CT AGN could be due to the suppression of AGN emission by absorption and its subsequent re-emission in the IR . The most reliable way of confirming the CT nature of an AGN is by X-ray spectroscopy . In a previous work , we presented the properties of X-ray detected DOGs by making use of the deepest X-ray observations available at that time , the 2Ms observations of the Chandra deep fields , the Chandra Deep Field North ( CDF-N ) , and the Chandra Deep Field South ( CDF-S ) . In that work , we only found a moderate percentage ( < 50 % ) of CT AGN among the DOGs sample . However , we pointed out that the limited photon statistics for most of the sources in the sample did not allow us to strongly constrain this number . In this paper , we further explore the properties of the sample of DOGs in the CDF-S presented in that work by using not only a deeper 6Ms Chandra survey of the CDF-S , but also by combining these data with the 3Ms XMM-Newton survey of the CDF-S. We also take advantage of the great coverage of the CDF-S region from the UV to the far-IR to fit the spectral energy distributions ( SEDs ) of our sources . Out of the 14 AGN composing our sample , 9 are highly absorbed ( N _ { H } > 10 ^ { 23 } cm ^ { -2 } ) , whereas 2 look unabsorbed , and the other 3 are only moderately absorbed . Among the highly absorbed AGN , we find that only three could be considered CT AGN . In only one of these three cases , we detect a strong Fe K \alpha emission line ; the source is already classified as a CT AGN with Chandra data in a previous work . Here we confirm its CT nature by combining Chandra and XMM-Newton data . For the other two CT candidates , the non-detection of the line could be because of the low number of counts in their X-ray spectra , but their location in the L _ { 2 - 10 keV } /L _ { 12 \mu m } plot supports their CT classification . Although a higher number of CT sources could be hidden among the X-ray undetected DOGs , our results indicate that DOGs could be as well composed of only a fraction of CT AGN plus a number of moderate to highly absorbed AGN , as previously suggested . From our study of the X-ray undetected DOGs in the CDF-S , we estimate a percentage between 13 and 44 % of CT AGN among the whole population of DOGs .