Let's see. I've Landau-Lifshitz vol. 2, fourth edition in the reprinted version of 1996 (English version) as well as the corresponding German version, which are the same concerning the sign conventions in GR. Note that earlier editions of this book had different sign conventions (see the table in MTW
@PeterDonis mentioned above).
Landau Lifshitz (4th edition)
The signature of the metric is west-coast: ##\eta_{\mu \nu}=\mathrm{diag}(1,-1,-1,-1)## for flat spacetime wrt. a Lorentzian basis.
The Riemann curvature tensor is defined by
$$\nabla_{\mu} \nabla_{\nu} A^{\rho}-\nabla_{\nu} \nabla_{\mu} A^{\rho}=A^{\sigma} {R^{\rho}}_{\sigma \mu \nu}$$
and the Ricci tensor by
$$R_{\mu \nu} = g^{\rho \sigma} R_{\rho \mu \sigma \nu}.$$
The Einstein field equations read
$$R_{\mu \nu}-\frac{1}{2} R g_{\mu \nu}=\frac{8 \pi G}{c^4} T_{\mu \nu}.$$
Chandrasekhar, The mathematical theory of black holes (1983)
The signature of the metric is west-coast: ##\eta_{\mu \nu}=\mathrm{diag}(1,-1,-1,-1)## for flat spacetime wrt. a Lorentzian basis.
The Riemann curvature tensor is defined by
$$\nabla_{\mu} \nabla_{\nu} A^{\rho}-\nabla_{\nu} \nabla_{\mu} A^{\rho}=-A^{\sigma} {R^{\rho}}_{\sigma \mu \nu}$$
and the Ricci tensor by
$$R_{\mu \nu} = g^{\rho \sigma} R_{\rho \mu \sigma \nu}.$$
The Einstein field equations read
$$R_{\mu \nu}-\frac{1}{2} R g_{\mu \nu}=-\frac{8 \pi G}{c^4} T_{\mu \nu}.$$
So the difference in sign is due to the definition of the Riemann curvature tensor (note that the Riemann tensor is antisymmtric wrt. to the 1st and 2nd as well the 3rd and 4th index), and the EFEs are the same in both books.
I'd be very surprised, if there were different sign conventions for the energy-momentum tensor of matter and radiation, because usually one defines energy densities as positive, ##T^{00}>0##.