Is omitting the pronoun the second time it appears in a sentence
I doubt any AmE or BrE speakers would consider this type of ellipsis to be incorrect provided clarity is left intact.
"I am in trouble, and need help."
See answers at:
When is it correct to repeat the subject?
Is it always necessary to repeat the pronoun before each verb?
More interesting might be why your friends particularly objected to the ellipsis of the subject pronoun "I" in your particular example sentence.
Even more interesting to me is why eliding "I" after conjunction "and" sounds fine whereas eliding it after conjunction "so" sounds strange to me.
"so" and "and" are both coordinating conjunctions.
1. Together with or along with; in addition to; as well as.
American Heritage Dictionary
1.along with; in addition to
Collins English Dictionary
"so" is a conjunction that expresses consequence or result of a cause. Usually in the style of "(cause/reason), so "result/consequence. Example:
"I blew up my boss's office, so I was fired."
Your question of whether you can elide the personal pronoun "I" from the second clause has really got me stuck.
"I blew up my boss's office, so was fired." (sounds like an unnatural
In the ELL answer I provided it says:
Since there is a coordinating conjunction between the two sentences,
leaving out the pronoun is fine.
Subject pronoun ellipsis question
However it's nowhere as simple as that.
I managed to find a quote from Practical English Usage here in this forum thread:
Ellipsis is not normally possible after other conjunctions besides
and, but and or.
However I don't find this to be a helpful rule either. To omit pronoun "I" after conjunction "so" sounds strange to me, but I don't know what the reason is. So I have made a list of sentences with ellipsis of the "I" pronoun. And here I am recognising conjunctions of different types as recognised on the Wikipedia article on conjunctions, ie., the distinction I make between coordinating conjunctions and subordinating conjunctions are as reflected in that article.
Using common coordinating conjunctions:
"I drink and (I) smoke." (sounds good)
"I drink, but (I) don't smoke." (sounds good)
"I either drink or (I) smoke." (sounds good)
"I drink, yet (I) don't smoke." (sounds good)
"I do not drink, nor (do I) smoke." (sounds acceptable)
"I do not drink, neither (do I) smoke." (sounds odd to me)
"I'm conscious about my health, so (I) do not smoke." (sounds odd to me)
Using common subordinating conjunctions:
"I drink after (I) smoke." (Bad)
"I drink before (I) smoke." (Bad)
"I volunteer because (I) care." (Bad)
"I am happy, since (I) am healthy." (Bad)
I can't think of a rule why omitting pronoun "I" after "so" isn't allowed or sounds strange. Hopefully someone else knows what the rule is.
I'd be interested to know if your friends who said your example was correct or wrong gave any reasons for their opinion.