Shared ground
These two verses assume that teaching and community life are tightly connected. A person can arrive claiming to be a legitimate Christian teacher, yet not bring “this teaching” (v.10). The letter treats that mismatch as serious enough to limit access and support.
The passage also treats certain kinds of hospitality as more than private kindness. Not receiving someone “into your house” and not offering a welcoming greeting are portrayed as ways to avoid giving a teacher credibility, resources, or a platform. The stated reason is that welcome can make the host a participant in the visitor’s “evil works” (v.11).
Where interpretation differs (only where needed)
What “this teaching” is. Many read it as the specific teaching emphasized in the letter’s warning about deceivers (see 2 John 7), especially confession about Jesus Christ. Others take it more broadly as the whole apostolic message the community has “heard from the beginning” (see 2 John 5–6), including how truth and love shape life together.
What “house” and “welcome” mean in practice. Some read “house” mainly as providing lodging and material support to traveling teachers, and “welcome” as a public greeting that signals approval. Others think “house” points to access to the house-church gathering itself, so the issue is granting a teaching venue, not merely a bed.
Why the disagreement exists
The letter is short and assumes a shared situation: traveling teachers, house-based meetings, and a known controversy. The phrases “this teaching,” “house,” and “welcome” are brief, so readers infer details from (1) the immediate context about deceivers, and (2) what hospitality meant socially in that world.
What this passage clearly contributes
Explicitly, the text says (1) some visitors do not bring “this teaching,” (2) such people are not to be received into the house or welcomed, and (3) welcoming links the host with the person’s evil works. The theological inference the passage invites is that endorsement and support are morally weighty actions: they can join a community to harmful teaching and its effects, even if the community is not the direct speaker of that teaching.