Context. GRB 211106A and GRB 211227A are two recent gamma-ray bursts (GRBs) whose initial X-ray position enabled us to possibly associate them with bright, low-redshift galaxies (z < 0.7). The prompt emission properties suggest that GRB 211106A is a genuine short-duration GRB and GRB 211227A is a short GRB with extended emission. Therefore, they are likely to be produced by a compact binary merger. However, a classification based solely on the prompt emission properties can be misleading. Aims. The possibility of having two short GRBs occurring in the local Universe makes them ideal targets for the search of associated kilonova (KN) emission and for detailed studies of the host galaxy properties. Methods. We carried out deep optical and near-infrared (NIR) follow-up with the ESO-VLT FORS2, HAWK-I, and MUSE instruments for GRB 211106A and with ESO-VLT FORS2 and X-shooter for GRB 211227A, starting from hours after the X-ray afterglow discovery up to days later. We performed photometric analysis to look for afterglow and KN emissions associated with the bursts, together with imaging and spectroscopic observations of the host galaxy candidates. We compared the results obtained from the optical/NIR observations with the available Swift X-Ray Telescope (XRT) and others high-energy data of both events. Results. For both GRBs we placed deep limits to the optical/NIR afterglow and KN emission. We identified their associated host galaxies, GRB 211106A at a photometric redshift z = 0.64, GRB 211227A at a spectroscopic z = 0.228. From MUSE and X-shooter spectra we derived the host galaxy properties, which turned out to be consistent with short GRBs typical hosts. We also compared the properties of GRB 211106A and GRB 211227A with those of the short GRBs belonging to the S-BAT4 sample, here extended up to December 2021, in order to further investigate the nature of these two bursts. Conclusions. Our study of the prompt and afterglow phase of the two GRBs, together with the analysis of their associated host galaxies, allows us to confirm the classification of GRB 211106A as a short GRB, and GRB 211227A as a short GRB with extended emission. The absence of an optical/NIR counterpart down to deep magnitude limits is likely due to high local extinction for GRB 211106A and a peculiarly faint kilonova for GRB 211227A.
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