peopleId <- LETTERS[1:9]

library(data.table)
DT <- data.table(giver = peopleId, mark_giver = FALSE, taker = peopleId, mark_taker = FALSE)

out <- matrix(NA_character_, nrow(DT), 2L)
for (i in 1L:nrow(DT)) {
  out[i, 1L] <- sample(DT[mark_giver == FALSE]$giver, 1L)
  out[i, 2L] <- sample(DT[mark_taker == FALSE & taker != out[i, 1L]]$taker, 1L)
  set(DT, match(out[i, 1L], peopleId), match("mark_giver", names(DT)), TRUE)
  set(DT, match(out[i, 2L], peopleId), match("mark_taker", names(DT)), TRUE)
}
out
#     [,1] [,2]
# [1,] "C"  "D"
# [2,] "H"  "G"
# [3,] "B"  "H"
# [4,] "A"  "E"
# [5,] "I"  "C"
# [6,] "G"  "I"
# [7,] "F"  "B"
# [8,] "E"  "F"
# [9,] "D"  "A"