Let's discuss the sentence by breaking it into 2 parts:
それはそうだかもしれませんが and 私(が/は)あなたの言うこと(は/が)通じます.
First Part
それはそうだかもしれませんが
-> Nothing wrong about particles. Good! You should say
それはそうかもしれませんが (removing だ and replacing り with れ.)
to sound more natural. You might find it strange that 知る turns into 知れ when connected to ません (negative,) but it is actually not 知る. It's "知れる."
Second Part
私はあなたの言うことが通じます
almost sounds natural!
Maybe you confused the first は with the one indicating subject.
Let's talk about this using the word わかる.
This 「は」 is originally 「に」. So 私にあなたの言うことはわかります。 is the beginning form. Note that the agent of the action is 私.
Then this 「に」 got 「は」 right after, and an emphasis has been put, which made 私にはあなたの言うことはわかります.
Then 「に」 got omitted. 私はあなたの言うことはわかります. Here, 私, which it the agent of the action, finally looks like a subject.
We got a problem of using 2 or more は in one sentence. So we change the latter は into が because 私 is the agent, which finally makes 私はあなたの言うことがわかります.
So, 私はあなたの言うことがわかります is natural.
But unlike the word わかる, I think it is weird to use 私は with the word 通じる.
When we say that "Somebody understand(s) my language (or words)," we do use the word 通じる. But it's the word that 通じる. Looks like the agent in this case is "the words!" We use it in the same way as わかる, like so:
言葉が 人に 通じる
So you should say:
あなたの言うこと(が/は) 私に 通じます.
And if you add は after に, which makes 私には, you put an emphasis on "私には" (=I can understand....)
So that being said, a little bit of reordering the words makes
私にはあなたの言うことが通じます。
But I don't think you can omit 「に」 in this case because "あなたの言うこと" looks like the agent.
P.S. you can talk about one's English skills as "xxxの英語."
So you can also say,
あなたの英語 は 通じます.
Note that there's a difference in nuance between using が and は.
This sentence ends in verb, so は puts an emphasis on the subject where が doesn't.
See this page for details.