There are several different ways to achieve OP's goal per this answer
(1) reorder_within() function to reorder name within period facets.
library(tidyverse)
library(forcats)
df <- data.frame(
  name = c("foo", "bar", "foo", "bar"),
  period = c("old", "old", "recent", "recent"),
  val = c(1.23, 2.17, 4.15, 3.65)
)
reorder_within <- function(x, by, within, fun = mean, sep = "___", ...) {
  new_x <- paste(x, within, sep = sep)
  stats::reorder(new_x, by, FUN = fun)
}
scale_x_reordered <- function(..., sep = "___") {
  reg <- paste0(sep, ".+$")
  ggplot2::scale_x_discrete(labels = function(x) gsub(reg, "", x), ...)
}
ggplot(df, aes(reorder_within(name, val, period), val)) +
  geom_col() +
  scale_x_reordered() +
  facet_grid(period ~ ., scales = "free", space = "free") +
  coord_flip() +
  theme_minimal() +
  theme(panel.grid.major.y = element_blank()) 

Or (2) similar idea 
### https://trinkerrstuff.wordpress.com/2016/12/23/ordering-categories-within-ggplot2-facets/
df %>% 
  mutate(name = reorder(name, val)) %>%
  group_by(period, name) %>% 
  arrange(desc(val)) %>% 
  ungroup() %>% 
  mutate(name = factor(paste(name, period, sep = "__"), 
                       levels = rev(paste(name, period, sep = "__")))) %>%
  ggplot(aes(name, val)) +
  geom_col() +
  facet_grid(period ~., scales = "free", space = 'free') +
  scale_x_discrete(labels = function(x) gsub("__.+$", "", x)) +
  coord_flip() +
  theme_minimal() +
  theme(panel.grid.major.y = element_blank()) + 
  theme(axis.ticks.y = element_blank())

Or (3) orders the entire data frame, and also orders the categories (period) within each facet group!
  ### https://drsimonj.svbtle.com/ordering-categories-within-ggplot2-facets
  # 
  df2 <- df %>% 
  # 1. Remove any grouping
  ungroup() %>% 
  # 2. Arrange by
  #   i.  facet group (period)
  #   ii. value (val)
  arrange(period, val) %>%
  # 3. Add order column of row numbers
  mutate(order = row_number())
df2
#>   name period  val order
#> 1  foo    old 1.23     1
#> 2  bar    old 2.17     2
#> 3  bar recent 3.65     3
#> 4  foo recent 4.15     4
ggplot(df2, aes(order, val)) +
  geom_col() +
  facet_grid(period ~ ., scales = "free", space = "free") +
  coord_flip() +
  theme_minimal() +
  theme(panel.grid.major.y = element_blank()) 

# To finish we need to replace the numeric values on each x-axis 
# with the appropriate labels
ggplot(df2, aes(order, val)) +
  geom_col() +
  scale_x_continuous(
    breaks = df2$order,
    labels = df2$name) +
  # scale_y_continuous(expand = c(0, 0)) +
  facet_grid(period ~ ., scales = "free", space = "free") +
  coord_flip() +
  theme_minimal() +
  theme(panel.grid.major.y = element_blank()) + 
  theme(legend.position = "bottom",
        axis.ticks.y = element_blank())

Created on 2018-11-05 by the reprex package (v0.2.1.9000)