This is based on the League of Legends gameplay, as I'm not familiar with the others. Since you specifically mention a lane partner, you're going 2v2 in the lane (unless other mobas have 3v3 or other matchup).
Things to Consider
Ranged vs Melee: Unless the enemy melee has a good gap closer skill at low levels (to forcibly fight you at close range), you can be more aggressive as you stay further away and its easier to retreat.
Engaging same target: If you engage an enemy and your ally goes around fighting the other enemy champion or creeps - don't be aggressive, and try to focus on assisting him instead to focus down a single hero (even if its the lower damage tanky one)
Crowd Control (Enemy): Opponents have lots of useful CC to keep your side snared in tower range or in the middle of a damaging spell? Defense first, unless you have counters to deal with it.
Crowd Control (Ally): Does your partner time his snares right to catch fleeing heroes and/or set them up for your skill shots? He's probably more experienced in coordinating well and can support an aggressive playstyle.
Pushing the lane: A partner who unnecessarily pushes the lane with the enemy around, where they get into tower coverage, may be overly aggressive - unfortunately, you should prioritize your wellbeing and be prepared to see them die due to overextending.
Tower dive: Its not necessarily bad to try and kill the tower on each opportunity. A partner wanting to tower dive after forcing enemies to leave lane or make it a 2v1 is good, one who just charges ahead without strategy is doomed to fail.
Be more aggressive if ...
- Enemies are very unskilled
- Partner cooperates well with you
- You're familiar with ally's hero (this directly affects how well you coordinate with him)
- Hero choices favor you in lane
Be more defensive if ...
- Partner has low health (you may need to defend lane while he goes back to heal)
- Enemies have high burst damage (esp. if both heroes working together can get you from 100-0, and they haven't shown a total lack of skill so far)
- Partner charges ahead without considering the situation: You may need to let them die in a 1v2, sacrificing yourself as well is a bad idea
- You don't get along with the lane partner
Dynamically adjusting to the circumstances is hard, and unfortunately needs a lot of experience and trial & error. Unfortunately, even at a simplistic level there is no single 'middle ground' to follow just like there is no 'average usable build' of items for all classes. So its better to put in the effort so you can recognize different playstyles & skill levels, and adjust for them accordingly.