We have examined ROSAT soft X-ray observations of a complete, distance-limited sample of Seyfert and LINER galaxies. X-ray data are available for 46 out of 60 such objects which lie within a hemisphere of radius 18 Mpc. We have constructed radial profiles of the nuclear sources in order to characterize their spatial extent and, in some cases, to help constrain the amount of flux associated with a nuclear point source. PSPC data from ROSAT have been used to explore the spectral characteristics of the objects with sufficient numbers of detected counts. Based on the typical spectral parameters of these sources, we have estimated the luminosities of the weaker sources in the sample. We then explore the relationship between the soft X-ray and H alpha luminosities of the observed objects; these quantities are correlated for higher-luminosity AGNs. We find a weak correlation at low luminosities as well, and have used this relationship to predict L_X for the 14 objects in our sample that lack X-ray data. Using the results of the spatial and spectral analyses, we have compared the X-ray properties of Seyferts and LINERs, finding no striking differences between the two classes of objects. However, both types of objects often exhibit significant amounts of extended emission, which could minimize the appearance of differences in their nuclear properties. The soft X-ray characteristics of the type 1 and type 2 active galaxies in the sample are also discussed. We then compute the local X-ray volume emissivity of low-luminosity Seyferts and LINERs and investigate their contribution to the cosmic X-ray background. The 0.5-2.0 keV volume emissivity of 2.2e38 ergs/s/Mpc^3 we obtain for our sample suggests that low-luminosity AGNs produce at least 9% of the soft X-ray background.
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