This is one of these questions where the answer is different depending on exactly what the question is.
Simplest version
Suppose you had linear regression with all of your $X$s on $n+m$ people and $Y$ on a random sample of $n$ of them. Let $\sigma^2$ be the residual mean square for the true line. The variance of your regression coefficient estimates (call them $\hat\beta$) would be $n^{-1}E[X^TX]^{-1}\sigma^2$ if you just used the complete data.
If you did proper multiple imputation and it worked perfectly, your within-imputation variance would be $(n+m)^{-1}E[X^TX]^{-1}\sigma^2$. The between-imputation variance, however, would include a term like $n^{-1}E[X^TX]^{-1}\sigma^2$ for the posterior uncertainty in the model parameters for $Y|X$ in the imputation. You end up not gaining anything, at least in large samples.
As an example, (in mice)
XY<-MASS::mvrnorm(500, mu=c(0,0,0,0,0), Sigma=diag(5)+1) X<-XY[,-1] Yfull<-Y<-XY[,1] Y[1:200]<-NA mouse<-mice(cbind(X,Y),m=50) fits<-with(mouse, lm(Y~X))
> MIcombine(fits[[4]]) Multiple imputation results: MIcombine.default(fits[[4]]) results se (Intercept) -0.04420686 0.07213510 X1 0.23700481 0.05430861 X2 0.22520806 0.05612435 X3 0.27398454 0.05608058 X4 0.14665648 0.05527157 > coef(summary(lm(Y~X))) Estimate Std. Error t value Pr(>|t|) (Intercept) -0.03942651 0.06353258 -0.6205716 5.353607e-01 X1 0.24226806 0.05374903 4.5073938 9.474943e-06 X2 0.23016556 0.05500893 4.1841488 3.780295e-05 X3 0.26290128 0.05641605 4.6600442 4.792135e-06 X4 0.15634203 0.05704397 2.7407285 6.504224e-03
Variations
Suppose the subsample with $Y$ is not representative but is sampled based on $X$ (and the original $X$ is more representative). In that case you still don't gain anything if the model is correctly specified, but if it's misspecified you get better fit to the population by using all the $X$s
Suppose that you have some additional variables $Z$ in addition to $X$ and $Y$ (and that you don't want $Z$ in your analysis model). In that case, using $(Z,X)$ to predict $Y$ will give you genuine additional information and better estimates of $Y|X$
Suppose you know (or want to assume) that structures in $X$ are aligned with the conditional distribution of $Y|X$. For example, suppose $X$ is a mixture distribution and you think the relationship between $Y$ and $X$ will be different for the components of the mixture. Or suppose you think that $Y$ is likely to be related to the first few principal components of $X$. There is no mathematical reason for these to be true, but there might well be domain reasons (eg if the first few principal components of your genetic data measure ancestry). In this setting, using the full data to learn about the structures in $X$ will let you partition the points with known $Y$ into their mixture components or let you estimate the first few principal components more accurately and so can be used to get better inference. This is semi-supervised inference in machine learning.
