The reasons the North would not have declared ware are pretty obvious, declaring war would effectively have recognized the CSA as a sovereign nation, which would have been counter to the aim of suppressing a rebellion and which would have opened the doors to other nations to recognize the CSA.
But why didn't the CSA declare war on the USA? That would have seemed to have been consistent with their claim to sovereignty, let alone the fact that the large war was is major endeavor at the time.
(Hope this hasn't been asked before, I did a search and couldn't find where it had been).
Lincoln was aware that the blockading of ports was an act of war. In fact, since an act of war is, by implication, taken against another state, some in his cabinet argued that a blockade would constitute a tacit recognition of the sovereignty of the Confederacy, something the North wanted to avoid. Lincoln was less interested in the legal definitions than in the military utility of the plan, and he approved it despite the objections.
On April 19, Lincoln issued his proclamation blockading Southern ports. It provided that “a competent force will be posted so as to prevent entrance and exit of vessels” from the ports of the states in rebellion. Then, to make the proclamation official, he signed this document, authorizing “the Secretary of State to affix the Seal of the United States to a Proclamation setting on foot a Blockade of the ports of the States of South Carolina, Georgia, Alabama, Florida, Mississippi, Louisiana, and Texas.” The seal was affixed to the blockade proclamation, which was announced that day. It was a de facto declaration of war by the Union against the Confederacy.
After the war, the Supreme Court issued an opinion fixing the exact dates on which the war began and ended. It held: “…The proclamation of intended blockade by the President may therefore be assumed as marking the first of these dates, and the proclamation that the war had closed, as marking the second.” Thus, according to the Supreme Court, Lincoln’s signature on this order sealing the imposition of the blockade marked the official beginning of the Civil War.
Now that that's settled out of the way regarding the Union's declaration of war towards the Confederacy, to answer the question towards the Confederacy's perspective.
To answer this question, we need to understand why the Confederates attacked Fort Sumter. This is mainly because of two factors: Jefferson Davis's pride on not appearing weak by allowing a federal fort in his "sovereign" territory, and also because quite literally; at least by the Confederate's perspective, Fort Sumter was on their land.
These aspects lead into the knowledge in that the idea of secession is a form of declaration of war itself for the Confederates because it lays active claim towards formerly federal land in South Carolina.
So in essence, the Confederates did not have a formal declaration of war because from their perspective they were already a legitimate state and it was not necessary to declare war against an adversary that was already impeding on their "sovereign" soil. The same way that the Union did not find it necessary to have a formal declaration of war against the Confederacy.