We present panchromatic observations and modeling of calcium-strong supernovae (SNe) 2021gno in the star-forming host-galaxy NGC 4165 and 2021inl in the outskirts of elliptical galaxy NGC 4923, both monitored through the Young Supernova Experiment transient survey. The light curves of both, SNe show two peaks, the former peak being derived from shock cooling emission (SCE) and/or shock interaction with circumstellar material (CSM). The primary peak in SN 2021gno is coincident with luminous, rapidly decaying X-ray emission (L-x = 5 x 10(41) erg s(-1)) detected by Swift-XRT at delta t = 1 day after explosion, this observation being the second-ever detection of X-rays from a calcium-strong transient. We interpret the X-ray emission in the context of shock interaction with CSM that extends to r < 3 x 10(14) cm. Based on X-ray modeling, we calculate a CSM mass M-CSM = (0.3-1.6) x 10(-3) M-circle dot and density n = (1-4) x 10(10) cm(-3). Radio nondetections indicate a low-density environment at larger radii (r > 10(16) cm) and mass-loss rate of M < 10(-4) M-circle dot yr(-1). SCE modeling of both primary light-curve peaks indicates an extended-progenitor envelope mass M-e = 0.02-0.05 M-circle dot and radius R-e = 30-230 R-circle dot. The explosion properties suggest progenitor systems containing either a low-mass massive star or a white dwarf (WD), the former being unlikely given the lack of local star formation. Furthermore, the environments of both SNe are consistent with low-mass hybrid He/C/O WD + C/O WD mergers.