  We report $\beta$-detected nuclear magnetic resonance ($\beta$-NMR)
measurements in Bi$_{2}$Se$_{3}$:Ca (BSC) and Bi$_{2}$Te$_{3}$:Mn (BTM) single
crystals using $^{8}$Li$^{+}$ implanted to depths on the order of 100 nm. Above
$\sim 200$ K, spin-lattice relaxation (SLR) reveals diffusion of
$^{8}$Li$^{+}$, with activation energies of $\sim 0.4$ eV ($\sim 0.2$ eV) in
BSC (BTM). At lower temperatures, the nuclear magnetic resonance (NMR)
properties are those of a heavily doped semiconductor in the metallic limit,
with Korringa relaxation and a small, negative, temperature-dependent Knight
shift in BSC. From this, we make a detailed comparison with the isostructural
tetradymite Bi$_{2}$Te$_{2}$Se (BTS) [McFadden et al., Phys Rev. B 99, 125201
(2019)]. In the magnetic BTM, the effects of the dilute Mn moments predominate,
but remarkably the $^{8}$Li signal is not wiped out through the magnetic
transition at 13 K, with a prominent critical peak in the SLR that is
suppressed in a high applied field. This detailed characterization of the
$^{8}$Li NMR response is an important step towards using depth-resolved
$\beta$-NMR to study the low-energy properties of the chiral topological
surface state (TSS). With the bulk NMR response now established in several
Bi$_{2}$Ch$_{3}$ tetradymite topological insulators (TIs), the prospect of
directly probing their chiral TSS using the depth resolution afforded by
$\beta$-NMR remains strong.
