At first hearing there seems to be a conflict. We are to "bear one another's burdens", yet, "each one shall bear his own load". Christian maturity desires to grasp the distinction and to apply it wisely.
The "burdens" (v. 2) that weigh so heavily upon a fellow believer are related to sin, whether his own or someone else's. A man "overtaken" or "caught" in any "trespass" (v. 1) is weighed down by his own guilt. We treat such situations "in a spirit of gentleness" because we too may fall under the same temptation, and that is how we would want to be considered and consoled. We must not overrate our own ability to avoid or overcome such temptations (v. 1,3).
Our Savior had to bear the sin of the world and atone for it on the cross through His blood (Heb. 9:11ff; 1 Jn. 1:7-9). Our heavy burden of guilt is removed only through the confession of sins and the Good News of salvation in Jesus (Prov. 28:13). This is the responsibility and the chief purpose of the office of the public ministry (Jn. 20:21-23). In addition, all believers can "fulfill the law of Christ" (Jn. 13:34) by forgiving in a private setting (Mt. 18:15ff; Jn. 5:16).
On the other hand, there is the personal "work" or "load" (v. 4,5) that constitutes the unique responsibilities of each believer's calling in life. This is the discipleship practiced by each member of Christ's Church as he lives in the world as a family member, a citizen, and a person with a particular role in society. Each Christian is bound by God's will (the Ten Commandments) to live life to the glory of God and the welfare of his neighbor (v. 10; Col. 3:17; Mt. 7:12; 22:37-40). That is his unique "load" and "work", to be carried out in faithfulness as a child of God (Tit. 3:8).
Neither of these two realms, of forgiving and comforting other sinners, or of living faithfully as God's child, can be carried out apart from our Lord Jesus and His Spirit.
Our foremost responsibility as God's children is that we trust in Jesus' saving work. We "examine" our faithfulness to God, confess our failings, and rely upon Christ. We cling to His Word and Sacraments because through them we have Him who bears us and all the "burdens" of this sinful life. He calls us who are "heavy laden" with the responsibilities of our discipleship "load" to find in Him the One who truly "bears" our "infirmities" and "sin" (Is. 53:4; Mt. 8:17; Jn. 1:29). and gives us "rest" (Mt. 11:28; Heb. 4:11-16).