Particle Emmiter on touch is not working. Why is my script not working?

I would check to see if a player is returned by the function GetPlayerFromCharacter before inserting the ParticleEmitters into their hands:

local Part = script.Parent

local function Speed(part)
    local parent = part.Parent
    if game.Players:GetPlayerFromCharacter(parent) then
	local right = Instance.new("ParticleEmitter",parent.RightHand)
	local left = Instance.new("ParticleEmitter",parent.LeftHand)
        right.Speed = 2
        left.Speed = 2
        right.RotSpeed = 10000
        left.RotSpeed = 10000
        right.Texture = ("http://www.roblox.com/asset/?id=3442350629")
        left.Texture = ("http://www.roblox.com/asset/?id=3442350629")
        right.Rate = 100
        left.Rate = 100
        right.Rotation = 555
        left.Rotation = 555
        right.Lifetime = 3
        left.Lifetime = 3
    end
end

Part.Touched:Connect(Speed)

The reason you should check to see if you're dealing with a player's character first is because it's possible that the part that touched your part is not a member of a player's character model, and then assuming that its parent contains a LeftHand and a RightHand would be unsafe.

Also, using connect is deprecated, meaning it's outdated and has a newer version, and probably won't work in the future. Use Connect instead.

As for your issue with the properties not changing, I would just make the ParticleEmitters in Studio first, put them in your script, and simply clone them and parent them to the character's hands.

local Part = script.Parent
local particleEmitter = script:WaitForChild("ParticleEmitter")

local function Speed(part)
	local parent = part.Parent
	if game.Players:GetPlayerFromCharacter(parent) then
local clone = particleEmitter:Clone()
		clone.Parent = parent.RightHand
		local clone2 = clone:Clone()
		clone2.Parent = parent.LeftHand
	end
end

Part.Touched:Connect(Speed)

And finally, to address your issue with multiple particle emitters being inserted into the player's hands:

local Part = script.Parent
local particleEmitter = script:WaitForChild("ParticleEmitter")

local function Speed(part)
	local parent = part.Parent
	if game.Players:GetPlayerFromCharacter(parent) then
		if not parent.RightHand:FindFirstChild("ParticleEmitter") then
			local clone = particleEmitter:Clone()
			clone.Parent = parent.RightHand
			clone.Enabled = true
		end
		if not parent.LeftHand:FindFirstChild("ParticleEmitter") then
			local clone = particleEmitter:Clone()
			clone.Parent = parent.LeftHand
			clone.Enabled = true
		end
	end
end

Part.Touched:Connect(Speed)

Just check to see if a particle emitter is already there.