Il me fait dire ça par un tiers.
Il me fait dire ça.
I'm so confused about this matter. The only difference is the addition of the "par" phrase, but are their meanings quite different grammatically?
My second attempt at an answer: To understand a causative faire sentence, sometimes we have to take into account the scheme of things that can go into it. Here's how we might proceed...
Faire has (1) an agent and (2) a recipient.
Dire has (3) an agent, plus any other complements for its normal usage. For example, dire usually takes (4) the thing said and (5) the person hearing it.
This structure is complex, and what makes it worse is that French marks some of those roles ambiguously. And not every sentence will have all of these things; some of them can be implicit.
Fortunately, some roles are easier to figure out. For example, (1) the agent of faire is simply the subject of the sentence ("il").
And your example includes ça, which is nice because the only role it can take is (4) the thing said.
Moreover, if we identify (2) the recipient of faire, then that same person is (3) the agent of dire (which is the key logic of a causative faire sentence).
So we need to identify just two more things: (2/3) the recipient of faire, and (5) the hearer of dire.
Here's where the problem comes in. Both of those roles can be expressed grammatically as an indirect object. That indirect object in your sentence is me.
So the question is: does me belong to faire or to dire ? That is, is me the person being made to talk, or the person who hears? (Naturally, we'll have to leave one role unstated either way.)
In your sentence, there's no grammatical way to decide. As Lambie writes below, there often is a semantic way to decide, i.e. the context will encourage a particular reading. And in any case, there are other ways to say this whole sentence that make it clearer.
But that's the unfortunate reality about the sentence. Note that the problem arises because a single item in the sentence could grammatically fulfill two different roles. Whenever that's true, we'll have the same problem. Some causative faire sentences are spared this issue.
Now let's see how "par un tiers" changes the situation.
It too can do one of two things: it can tell us (2/3) the recipient of faire/agent of dire, or it can tell us something entirely new: (6) how the thing is said.
If "par un tiers" is (2/3) the recipient of faire, that forces me to become (5) the person hearing. So in this sense it would solve our problem about me.
He makes a third party say it to me.
However, if "par un tiers" is (6) how the thing is said, then me is still undecided about the two roles mentioned above:
He has it said to me via a third party.
(Notice that this means more or less the same thing as "He makes a third party say it to me.")
He has me say it via a third party.
So that's about it. We're still left with ambiguity. But ambiguity comes from examining sentences in isolation. When you encounter it in the real world, hopefully the context makes it plain. :)