Question:
What if a person doesn’t go to another church but wants to break bread with us?

Answer:
They would have to know, believe, and desire to have part with what we stand for in order to have fellowship or “communion” with us.

First Corinthians 10:16, 17 gives us the doctrine regarding what the emblems stand for in the Lord’s Supper. Notice that the cup is mentioned first instead of the bread, as when the Lord instituted His supper (Luke 22:19, 20). The reason for this is because a person who takes these emblems must first have fellowship with the blood of Christ (in other words, he must be saved). If he is saved, then he can have fellowship with the bread, which speaks of the “one body” (v. 17) of Christ, which is composed of all who are saved. “The Lord added to the church daily such as should be [or, were being] saved” (Acts 2:47). So when we are gathered together as an assembly, we are to represent the “one body,” which is not just the brethren, but the whole body of believers.
Second Timothy 2:19-22 gives instruction for the “last days” we live in (2 Timothy 3:1), which is right before the Lord comes to take His church home to glory. Different teachings have come into the professing church, and we are told that we must be separate from both vessels to honor (saved) and dishonor (unsaved) who have “iniquity,” which is anything unscriptural, if we are going to be a vessel fit “for the Master’s use, and prepared unto every good work.”
We must know people in order to know if they are walking orderly or have false teaching. Fellowship or “communion” (1 Corinthians 10:17), is based on knowledge. The verse: “Lay hands suddenly on no man, neither be partaker of other man’s sins: keep thyself pure” (1 Timothy 5:22), has to do with the fellowship with believers, the body of Christ.
“Let a man examine himself, and so let him eat of that bread, and drink of that cup” (1 Corinthians 11:28), is not a message to those who are desiring to remember the Lord with the assembly for the first time, but it was written to believers already breaking bread (see who this epistle was written to: 1 Corinthians 1:2). Those who break bread in the assembly should examine, or judge themselves in the light of His presence before they eat of the Lord’s supper. They should examine: (1) The company they keep; (2) the habits they have; (3) their affections; (4) thoughts; and (5) motives.
We are told in 1 Corinthians 5:12 that the assembly is to judge those who are “within” the assembly, and this would include those who desire to come in to the assembly. It wouldn’t be orderly to bring them in, and then judge them and have to put them out! Three things that would keep someone out are: (1) doctrinal evil (Galatians 5:9); (2) moral evil (1 Corinthians 5:7, 13); and (3) evil associations (1 Corinthians 15:33).

The whole assembly must judge–not just one person, or a group of brothers. The “ye” in 1 Corinthians 5:4 and 12 is the local assembly, for the apostle is writing to “the church of God which is at Corinth” (1 Corinthians 1:2). Sisters are a vital part of the assembly and they should have say in reception, and when other discipline in the assembly is needed. The Lord gives further instruction in Matthew 18:17, 18 on assembly discipline.