You should start with a linear regression with two columns of response, eg:
P1 ~ some prior
P2 ~ some prior
M_{det1} ~ Normal(R_{det1,P1} * P1 + R_{det1,P2} * P2, error_1)
M_{det2} ~ Normal(R_{det2,P1} * P1 + R_{det2,P2} * P2, error_2)
Then you can introduce more complex structure, such as the correlation between M_{det1} and M_{det2}:
cov ~ LKJ
mu1 = R_{det1,P1} * P1 + R_{det1,P2} * P2
mu2 = R_{det2,P1} * P1 + R_{det2,P2} * P2
[M_{det1}, M_{det2}] ~ MvNormal([mu1, mu2], cov)