Apologies in advance if you specifically want to compare plots, because I don't know about that. But, I assume you have a continuous dependent variable and either continuous or factorial predictor or maybe several predictors. In this case I would compare the two groups statistically by putting them into the same lmer model - e.g. (with one continuous predictor):
model<-lmer(outcome ~ (1|person_id) + group*predictor, data=data) #edited to remove code redundancy
"group" being a factorial variable with two levels, group1 and control. Then, if the group*predictor interaction is significant, you have evidence that outcome ~ predictor relationship is statistically different for group1 vs. control. Then, you can run the simple models of outcome ~ predictor + (1|person_id) separately for each group and report the fixed main effects separately for each group, and also plot the 2 slopes into the same figure.
Edited to respond to your comment:
I think you got an output from R looking like this:
Linear mixed model fit by REML. t-tests use Satterthwaite's
Formula: outcome ~ (1 | id) + group * predictor
REML criterion at convergence: 851.8
Min 1Q Median 3Q Max
-2.05211 -0.66622 -0.02345 0.67188 2.56591
Groups Name Variance Std.Dev.
id (Intercept) 0.004382 0.0662
Residual 0.967369 0.9835
Number of obs: 300, groups: id, 60
Estimate Std. Error df t value
(Intercept) 4.34483 0.32619 295.67804 13.320
group1 -0.84478 0.44085 295.96649 -1.916
predictor -0.08481 0.07839 287.76043 -1.082
group1:predictor 0.19004 0.10647 288.70985 1.785
(Intercept) <2e-16 ***
group1 0.0563 .
group1:predictor 0.0753 .
(this is made based on simulated data with 60 participants and 5 observations per participant. I have no significant effects there, but never mind that).
So, if you look at the "Fixed effects" part, "group1" effect is an effect comparing group 1 to the control group (estimate for "predictor" is the continuous predictor main effect, here non-significant and negative). Because "group" is a factorial effect, it compares one group to another. "Control" is the default to which "group1" is compared to. That's why it is not shown in the output. With a factorial predictor, you always see in the output the number of levels of the predictor minus 1 level - because that's the level the other levels are compared to.
In my mock data, the results suggest that outcome was lower in group1 than in control group, because the estimate is negative, but not significantly so. If your "group1" effect was significant and positive, it suggests that outcome was significantly higher for people in group1 than in control, and vice versa if it was significant and negative. You get estimated marginal means for "control" and "group1" from emmeans package:
em<-emmeans(model, specs = pairwise ~ group)
Or, you can of course check and plot raw means by group too.
Of course, if the interaction is significant, you should focus on that more than on "group" main effect. You can for instance plot separate slopes of predictor for each group by using:
preds<-ggpredict(model, c("predictor", "group"))
plot1<-ggplot(preds)+geom_line(aes(x=x, y=predicted, color=group))
plot2<-plot1+scale_color_discrete(name="Group", labels=c("Control", "Group1"))
plot3<-plot2+labs(x="Predictor", y="Outcome", title="title")