Learn English – “Been seeing somebody doing something” vs “Saw somebody has been doing something”

Your first sentence is:

I've been seeing him posting stuff on his blog

The word see has many possible meanings: among these are

• you observe something
• you are aware that something is happening
• you meet with somebody

When see is followed by a thing, the only valid meaning is observe.

When see is followed by a person, two of these meanings are valid- observe and meet with.

When see is followed by a that-clause, the meaning be/become aware of are is valid.

If you are watching somebody, you can say "I see him", but people do not usually say "I am seeing him", because the use of a continuous tense of "see" implies the "meet with" meaning. So, you can say

I saw him in the supermarket yesterday - observation

I have been seeing my analyst for three years - meaning regular meetings

I have been seeing John in Accounts - meaning you have been dating him (another form of regular meetings).

If you want to use a continuous tense meaning observing, you have to use "watch":

I've been watching him posting stuff on his blog

This means that you were sitting next to him for a while, and he was posting stuff on his blog.

Moving on to your second sentence:

I saw he's been posting stuff on his blog

If you observed him, you can't say "I saw he." you have to say "I saw him", because what you saw must be the object. For example,

I saw him posting stuff on his blog.

When you say "I saw he...", it's really a that-clause where the that has been omitted. The omission is normal: you can tell that it's happened because the pronoun is a subject-pronoun. The complete sentence is therefore:

I saw that he's been posting stuff on his blog

With a that-clause, the appropriate meaning is to "be/become aware of". The first part of the sentence therefore means that, at some time in the past, you became aware that he had been posting stuff on his blog. It's perfect simple- a completed action- so you aren't aware of posts since you last looked: the posting (as far as you are aware) finished before you became aware of it. you therefore should use past perfect continuous:

I saw that he had been posting stuff on his blog.

If you are aware now that he is still posting on his blog, you should use the present simple:

I see that he has been posting stuff on his blog.

If you have been keeping an eye on his blog for a while and are still doing so, you can use present perfect simple for "see":

I have seen that he has been posting stuff on his blog.