This work now:

library(shiny)
library(dplyr)
library(ggplot2)

innerModUI <- function(id) {
    tagList(
        uiOutput(NS(id, "inner_slider")),
        plotOutput(NS(id, "inner_plot"))
    )
}

innerMod <- function(id) {
    moduleServer(id, function(input, output, session) {
        mpg_min <- round(min(mtcars$mpg, na.rm = TRUE))
        mpg_max <- round(max(mtcars$mpg, na.rm = TRUE))

        output$inner_slider <- renderUI({
            sliderInput(session$ns("slider2"), label = "inner module slider", min = mpg_min,
                        max = mpg_max, value = c(mpg_min, mpg_max), step = 1)
        })

        output$inner_plot <- renderPlot({
            req(input$slider2)
            data <- dplyr::filter(mtcars, dplyr::between(mpg, input$slider2[1], input$slider2[2]))
            ggplot(data, aes(mpg, wt)) + geom_point()
        })
    })
}

outerModUI <- function(id) {
    tagList(
        uiOutput(NS(id, "outer_slider")),
        plotOutput(NS(id, "outer_plot")),
        innerModUI(NS(id, "inner"))
    )
}

outerMod <- function(id) {
    moduleServer(id, function(input, output, session) {
        innerMod("inner")

        mpg_min <- round(min(mtcars$mpg, na.rm = TRUE))
        mpg_max <- round(max(mtcars$mpg, na.rm = TRUE))

        output$outer_slider <- renderUI({
            sliderInput(session$ns("slider1"), label = "outer module slider", min = mpg_min,
                        max = mpg_max, value = c(mpg_min, mpg_max), step = 1)
        })

        output$outer_plot <- renderPlot({
            req(input$slider1)
            data <- dplyr::filter(mtcars, dplyr::between(mpg, input$slider1[1], input$slider1[2]))
            ggplot(data, aes(mpg, wt)) + geom_point()
        })
    })
}

ui <- fluidPage(
    fluidRow(
        outerModUI("outer")
    )
)

server <- function(input, output, session) {
    outerMod("outer")
}

shinyApp(ui = ui, server = server)