To achieve the required recycling rates, organic recycling via composting should be widely introduced in Poland for selectively collected biowaste. However, this process not only produces compost but also leachate (LCB), a nitrogen- and organics-rich liquid by-product. So far there has been limited information on the application of anaerobic digestion (AD) for treating LCB, which has fermentative potential. However, for effective methane production (MP) via AD, the ratio of chemical oxygen demand to total Kjeldahl nitrogen (COD/TKN) and pH of LCB are too low; thus, it should be co-digested with other organics-rich waste, e.g., glycerine (G). The present study tested the effect of G content in feedstock (in the range of 3–5% (v/v)) on the effectiveness of co-digestion with LCB, based on MP and the removal of COD. MP was accessed by using an automatic methane potential test system (AMPTS). Regardless of the feedstock composition (LCB, or LCB with G), the efficiency of COD removal was over 91%. Co-digestion not only increased MP by 6–15%, but also the methane content in the biogas by 4–14% compared to LCB only (353 NL/kg CODadded, 55%). MP and COD removal proceeded in two phases. During co-digestion in the 1st phase, volatile fatty acids (VFA) accumulated up to 2800 mg/L and the pH decreased below 6.8. The presence of G altered the shares of individual VFA and promoted the accumulation of propionic acid in contrast to LCB only, where caproic acid predominated. An initial accumulation of propionic acid and acidification in the mixtures decreased the kinetic constants of MP (from 0.79 to 0.54 d−1) and the rate of COD removal (from 2193 to 1603 mg/(L·d)). In the 2nd phase, the pH recovered, VFA concentrations decreased, and MP was no longer limited by these factors. However, it should be noted that excessive amounts of G, especially in reactors with constant feeding, may cause VFA accumulation to a greater extent and create a toxic environment for methanogens, inhibiting biogas production. In contrast, digestion of LCB only may lead to ammonium buildup if the COD/TKN ratio of the feedstock is too low. Despite these limitations, the use of AD in the treatment of LCB as a sustainable “closed-loop nutrient” technology closes the loop in composting of biowaste.