Student pilot with instructor reported a NMAC during cruise requiring evasive action to avoid a collision.

Narrative

At XA:10 Zulu; my instructor and I departed the airport in Aircraft X from Runway 22. At XA:13 Zulu; Aircraft Y departed Runway 33. Roughly 3 miles off the departure end of the runway; I called that we were climbing to 3;500' for an east departure. Shortly after this; Aircraft Y called that they were departing to the south; this was the last we heard from the aircraft. Once on course; heading East and at 3500'; I noticed Aircraft Y at the same altitude heading southeast; roughly XA:15 Zulu. At this time; I was going to call Aircraft Y over the CTAF to ask their intentions; as we were at the same altitude. I was under the hood during this time paying attention more to my airspeed and altitude but was well aware that Aircraft Y was at the same altitude heading our way. My instructor then made a call asking their intentions; and they had no response but descended to 3000. We started climbing to increase our distance from them as we passed over top. Out of no where Aircraft Y began climbing again heading straight for us. My instructor then took controls and began a steep climbing left turn. We then had the aircraft in sight off to my left; I had my hood off at this point to help view with my instructor what they were doing. They buzzed right in front of us at the exact location we would have been had we not climbed at a significant rate and made a left turn. We ended up being around 3800 to 3900 feet MSL at the intersection point while they were at 3500 feet MSL. I will never again trust that a plane knows our location and will always keep an eye on my G-1000 instrument making sure no one is near and that this situation never happens again.