1-President x said on Tuesday that there is an abundant opportunity for investment in the energy sector, noting we can fulfill a part of Europe’s energy needs.
Noting is a participle which modifies the clause "we can fulfill a part of Europe’s energy needs."
2-He announced his decision on Saturday in a letter to senior members of his Cambodia National Rescue Party that was published on social media, saying he was standing aside "for the sake of the party".
Saying performs the same function that *noting *does in the first sentence , modifying. "he was standing aside 'for the sake of the party'."
Notes and says are verbs,and, cannot be exchanged for the participles. They could be used by adding a conjunction and using the tense of the first clause for time agreement:
1-President x said on Tuesday that there is an abundant opportunity for investment in the energy sector, and noted we can fulfill a part of Europe’s energy needs.
2-He announced his decision on Saturday in a letter to senior members of his Cambodia National Rescue Party that was published on social media, and said he was standing aside "for the sake of the party".
It would not be correct to use "by noting" or"by saying", as that would mean that the action in the first clause was accomplished by the action in the second clause. That would not be the case, completely, in either sentence..