Testing simple mediation model with two separate outcome variables I'd like to test whether a variable mediates the relationship between an IV and 2 DVs. The DVs will likely be highly correlated, as they're essentially measuring opposite constructs. Should I do this using 2 separate regressions (one for each DV) or should I use multivariate regression? 
I don't know how to do SEM and the variables aren't latent. 
 A: SEM is actually a good approach for this kind of analysis (see this question for some ideas why). In a nutshell, even though you don't think you need to analyze latent variables (though your opposing DVs sound like they could be manifestations of one?...), SEM is still helpful because of the ability to simultaneously model multiple DVs, and to assess the impact of particular model constraints. 
I appreciate that you are unfamiliar with SEM, so here is some example code to get you started, which I've amended from the lavaan tutorial on mediation. 
#Simulate some example data for X, M, and two DVs (Y1 and Y2)
set.seed(1234)
X = rnorm(100)
M = 0.5*X + rnorm(100)
Y1 = 0.7*M + rnorm(100)
Y2 = -0.6*M + rnorm(100)
Data <- data.frame(X = X, Y1 = Y1, Y2 = Y2, M = M)

#Install and call lavaan package for SEM
install.packages("lavaan")
library(lavaan)

#Specify simple mediation model of Y1 and Y2 on X through M
model <- ' 
# direct effect
Y1 ~ c1*X
Y2 ~ c2*X

# mediator
M ~ a*X
Y1 ~ b1*M
Y2 ~ b2*M

# indirect effects (a*b)
ab1 := a*b1
ab2 := a*b2

# total effect
total1 := c1 + (a*b1)
total2 := c2 + (a*b2)
'
#Fit model to simulated data, and request bootstrapped estimates
fit <- sem(model, data = Data, se = "bootstrap")

#Request summary output of parameter estimates
summary(fit)

You can run all of that syntax in R to have a complete working example. Then, you could simply load your data into R, and replace variable names for X, M, and your DVs, and run the code above for your actual model. The output is very similar--interpreted identically--to your normal regression output (the parameter estimates are unstandardized). 
Taking further advantage of SEM capabilities, you could also consider fitting a model in which you constrain the b paths (Y1 ~ M, Y2 ~ M) to be equal in magnitude and see whether that significantly degrades the fit of the model. In this way, you could test for which outcome (Y1 or Y2) M is a stronger mediator. 
A: I like @jsakaluk approach, and I would like to add another point that you may consider trying. The type of question that you have (mediation, in which DVs could be highly correlated) in a very common issue in both psychometrics and econometrics. One way to deal with this is to conduct a so-called robustness analysis. It is done in a couple of simple steps (one example of this approach is Tokarev, Phillips, Hughes & Irwing, 2017). I will provide a non-technical explanation and logic for this approach below. 
Step 1
Start with your mediation model, for example, DV1 and DV2 -> Mediator -> Outcome variable. 
Important note: For simplicity, I only present here a so-called full mediation model, in which the effect of your DV1 and DV2 on the outcome variable goes only through Mediator, so there are no individual direct effects from each predictor DV1 and DV2 on the outcome variable. If there were both direct effects and mediated effects, you would typically call this partial mediation. Please, think carefully if direct effects are also part of your model. Some useful discussion on partial vs full mediation is provided in a recent Mediation review by Memon, Cheah, Ramayah, Ting, and Chuah (2018). 
Examine the parameters of this model. @jsakaluk provided R code, but if you are working with Mplus, I can add the code for you (feel free to ask). With highly correlated DV1 and DV2, you would expect some of the effects in your model to be statistically non-significant and/or small in magnitude. For example, suppose
DV1-> Mediator is statistically significant and has expected magnitude, but DV2 -> Mediator is non-significant and/or has a very small magnitude. 
At this point, it would be premature to conclude that there is really no effect of DV2 on Mediator (and thus no mediation for: DV2 -> Mediation  -> Outcome variable). Instead, you may note that such an effect could have resulted from high multicollinearity between DV1 and DV2. Therefore, to test this possibility, you are going to do the following. 
Step 2
Run two individual mediation models by separating correlated predictors. 
a) DV1 -> Mediator -> Outcome variable 
b) DV2 -> Mediator -> Outcome variable
Assess the coefficients of the two models, and usually, you would expect a very different result for model b) and specifically for the DV2 -> Mediator path. This will allow you to test the effect of DV2 on Mediator more accurately since you are circumventing the issue of multicollinearity presented previously.  
If you end up with both mediated models a) and b), this will give you a good opportunity to compare the effects of both mediated models (e.g. which mediation is stronger). 
Final note: a good critical overview of Mediation is provided in Zhao, Lynch and Chen, Q. (2010). You may quickly read it to get familiar with some critical issues in mediation analysis such as power. 
References
Memon, M. A., Cheah, J., Ramayah, T., Ting, H., & Chuah, F. (2018). Mediation Analysis Issues and Recommendations. Journal of Applied Structural Equation Modeling, 2(1), 1-9.
Tokarev, A., Phillips, A. R., Hughes, D. J., & Irwing, P. (2017). Leader dark traits, workplace bullying, and employee depression: Exploring mediation and the role of the dark core. Journal of Abnormal Psychology, 126(7), 911-920.
Zhao, X., Lynch Jr, J. G., & Chen, Q. (2010). Reconsidering Baron and Kenny: Myths and truths about mediation analysis. Journal of Consumer Research, 37(2), 197-206. 
