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Also, it is okay to have two にs in the sentence because they are different にs. 子供に marks it as a relational/indirect object. 買いに marks purpose of action. (Also take a look at sawa's answer to another question)

Also, it is okay to have two にs in the sentence because they are different にs. 子供に marks it as a relational/indirect object. 買いに marks purpose of action. (Also take a look at sawa's answer to another question)

[c'] 彼に私のこと｛について・にかんして｝分からせます。
[d'] 彼を私のこと｛について・にかんして｝分からせます。

[c'] 彼に私のこと​｛について・にかんして｝分からせます。
[d'] 彼を私のこと​｛について・にかんして｝分からせます。

Aha! There seems to be no difference! But there is. The difference between を and に lies in the agent/causee's volition.

When に is used, the causee does the action in line with his volition.

When を is used, it is independent of the causee's volition. This means it can have the nuance of being forced. (Although may not always be the case, it depends on context)

So 2[b] means that "you" intended to sit there, and that person let you. 2[a] can also mean the same thing, but it can also mean that that person forced you to sit there.

EDIT: (My intuition says that そこに座らせます　is fine, but 私にそこに座らせます is not. I think it should be 私にそこで座らせます instead, or else I'd have two にs performing "indirect object" role. Please someone let me know in the comments if I got this portion right/wrong)

As in (2), choosing to use に or を can mean different things. The exception to this rule is that you cannot use を more than once in a clause.

EDIT: (My intuition says that そこに座らせます　is fine, but 私にそこに座らせます is not. I think it should be 私にそこで座らせます instead, or else I'd have two にs performing "indirect object" role. Please someone let me know in the comments if I got this portion right/wrong) 

Yes 私を has to be used if you want to maintain the use of そこに.

Choosing to use に or を can mean different things. The exception to this rule is that you cannot use を more than once in a clause.

The difference between を and に lies in the agent/causee's volition.

When に is used, the causee does the action in line with his volition.

When を is used, it is independent of the causee's volition. This means it can have the nuance of being forced. (Although may not always be the case, it depends on context)

So (2)[b] (if grammatical) means that "you" intended to sit there, and that person let you. (2)[a] can also mean the same thing, but it can also mean that that person forced you to sit there.