When I hear pastors saying Satan shoots "fiery darts of doubt or immorality" into our minds, it confuses me – Satan is not omnipresent and cannot inhabit a believer. How then can Satan shoot those thoughts? Is it not our own self who allows those thoughts?
The phrase you quoted comes from Paul’s writing in Ephesians 5:
The enemy cannot indwell a believer, according to Scripture, but he and his demons can still exert influence over a believer. He accomplishes his work through the agency of countless demons, so though Satan is not omnipresent, he can still make his presence felt everywhere.
Paul describes the opposition we face from Satan as “flaming arrows,” and your pastor suggested some forms that those “flaming arrows” may take, which the Bible supports. For example, the Bible says the enemy has the power to influence our thinking and actions by invoking fear (Hebrews 2:14-15) and temptation (Mark 1:13; 1Thess 3:5), and perpetuating spiritual ignorance (Luke 8:12). By enticing our flesh, the enemy can move our thoughts and feelings in ways that suit him, provided we are living in our flesh and not walking by the Spirit.
In Ephesians, Paul teaches that our faith in the Lord is a shield against these things. In other words, by our faith in Christ, we are born again with a new spirit that can resist the enemy and live according to the Holy Spirit’s direction. For more information on this passage, we recommend you listen to our Ephesians Bible study.