How would I make the dialogue GUI show locally for a single player and not the whole server?

there are multiple ways to accomplish the thing you want, but i am going to go with the easy way:

if you already have a placeholder dialogue gui in the player's playergui, you can fire a remote to only that client with the specific strings and other data you may need.

--[[the "replication" method, may be slow because you are doing .touched on the server but more reliable since exploiters are unable to fire touch interests on the client]]--
-- local
local remote = workspace.remote -- some location
remote.OnClientEvent:Connect(function(strings,...)
    -- "strings" can either be a table of strings or a single string
    local args = {...}
    -- do gui stuff here 
end)

-- server
local remote = workspace.remote
local someplatform = workspace.Part
someplatform.Touched:Connect(function(p)
    local plr = game:GetService("Players"):GetPlayerFromCharacter(p.Parent)
    remote:FireClient(plr,{"hi","what"},somethingidk)
end)

--[[the "client only" way, less reliable unless you know how to secure a .touched event]]--
local someplatform = workspace.Part
local plr = game:GetService("Players").LocalPlayer
someplatform.Touched:Connect(function(p)
    -- do gui stuff here
end)

keep in mind exploiters can use a basic function to fire all touch interests so i would use something like region3 but .touched still works well