How do you change a value with a local script?

Well, I would recommend you to do it on Server-Side to prevent any type of exploitation. This is what you might do :

local datastore = game:GetService("DataStoreService")
local ds1 = datastore:GetDataStore("CashSaveSystem")

game.Players.PlayerAdded:Connect(function(plr) -- Capital 'C'
    local folder = Instance.new("Folder", plr)
        folder.Name = "leaderstats"
        local cash = Instance.new("IntValue", folder)
        cash.Name = "Money"

        cash.Value = ds1:GetAsync(plr.UserId) or 500
        ds1:SetAsync(plr.UserId, cash.Value)

        cash.Changed:connect(function()
            ds1:SetAsync(plr.UserId, cash.Value)
        end)

    local function PayCheck() -- Creating a function
        local paycheck = workspace.Leaderboard.PayCheck
        local teams = game:GetService("Teams")

        local payCheckTable = {
            ["Citizen"] = 120,
                ["Criminal"] = 150,
                ["Inmate"] = 0,
                ["Police"] = 200,
            ["HeavyGunDealer"] = 225,
                ["GunDealer"] = 165,
                ["AdminOnDuty"] = 1000      
        } -- Thanks to Feelings_La

        -- Time for checking the player's team
        for i, v in pairs(payCheckTable) do -- i = team of the player, v = price of paycheck
            if plr.Team == team[i] then -- Checks if the player is in the team in the table
                paycheck.Value = v
            end
        end
    end

        while wait(60) do
        PayCheck() -- Triggers our function
            cash.Value = cash.Value + script.PayCheck.Value
        end

    -- Don't add code after a while loop, as it is never ending. So, the code after the while loop never gets checked
end)

Lemme know if it helps!

You are changing your value locally not globally.

Use Remote Function or Events. They are great to allow you communicate between server and client! Becareful when you are using remote function/events though. Since exploiters can manipulate it.

More info about Remote Function/Events:

On your local script: I'd suggest you to make a table for paycheck instead of messy giant script.

local player = game.Players.LocalPlayer
local leaderstats = player:WaitForChild("leaderstats")
local leaderboard = game.Workspace.Leaderboard


local ReplicatedStorage = game:GetService("ReplicatedStorage")
local UpdateBoard = ReplicatedStorage.UpdateBoard -- create a new remote function

local payChecks = {
    ["Citizen"] = 120,
    ["Criminal"] = 150,
    ["Inmate"] = 0,
    ["Police"] = 200,
    ["HeavyGunDealer"] = 225,
    ["GunDealer"] = 165,
    ["AdminOnDuty"] = 10000

}

for team, paycheck in pairs(payChecks) do
    if player.Team == game:GetService("Teams")[team] then
        leaderboard.PayCheck.Value = paycheck
    end
end

while wait(60) do -- DO THIS on local script instead of server!
    UpdateBoard:InvokeServer(leaderboard.PayCheck.Value)
end

Then on your server script:

-- THIS IS FOR SAFETY REASON
local payChecks = { 
    ["Citizen"] = 120,
    ["Criminal"] = 150,
    ["Inmate"] = 0,
    ["Police"] = 200,
    ["HeavyGunDealer"] = 225,
    ["GunDealer"] = 165,
    ["AdminOnDuty"] = 10000

}
local RS = game:GetService("ReplicatedStorage")
local UpdateBoard = ReplicatedStorage.UpdateBoard

UpdateBoard.OnServerInvoke = function(plr, payCheck)
    if payCheck == payChecks[player.Team] then
        plr.leaderstats.Cash.Value = plr.leaderstats.Cash.Value + payCheck
    end
end

you should use a remote event and do it from a server script cus if u change the value in a local script it wont register to the server and it will only save the data thats on the server so you'll need a server script to change the value