Does sin hinder one's ability to speak God's word? A friend told me that "bad people" are limited when it comes to spreading the Word of God.

According to scripture, how, and in what way, does sin hinder one's ability to spread the Word of God?
Your question is a little more complicated than one might think at first blush!

"Bad people" are the operative words, in my opinion. Bad people may attempt to preach and teach God's Word, but they do so for base motives, which could include (but are not limited to) the following:


money
fame
power
lust for any of the above, and/or lust for sexual immorality, such as the "man of God" who takes advantage of his female followers


The Bible calls bad men (and bad women) who attempt to preach and teach God's Word with their own particular and sinful slant "false teachers." The task of mature Christian believers, such as elders, deacons, pastors, ministers, priests, and so on, is to be on guard for false teachers and to protect the flock from these predators, whom Jesus called "ravenous wolves" in Matthew 7:15; 10:16; and Luke 10:3. 

The important word in the last sentence is, I suggest, mature. Spiritual maturity in a church leader means in part that he or she is able to confront and refute false teachers with sound doctrine and teaching. 


  "[Elders] must pay attention to the reliable message as it has been taught to them so that they can encourage people with healthy instruction and refute those who speak against it" (Titus 1:9 CEB).


Is it possible for an ungodly man or woman to benefit spiritually a hearer or two now and then? Yes, I believe so. The writer to the Hebrews tells us that God's word is  


  "living and active and sharper than any two-edged sword . . . [and] able to judge the thoughts and intentions of the heart" (Hebrews 4:12, excerpts, NASB Updated).


False teachers may even lead with truth (in the sense of starting with the truth), and then follow up with a partial truth or a distortion of the truth. The truth with which  they "lead" might be straight Scripture. The simple quoting of a Scripture could make all the positive difference in the life of a person, particularly if he or she extracts themselves from the false teacher soon enough! 

Interestingly, the apostle Paul had an interesting perspective on preachers and teachers who preach and teach with the wrong motives. Paul said, 


  "Some indeed preach Christ even of envy and strife; and some also of good will: the one [do it] of love, knowing that I am set for the defence of the gospel; but the other proclaim Christ of faction, not sincerely, thinking to raise up affliction for me in my bonds. What then? only that in every way, whether in pretence or in truth, Christ is proclaimed; and therein I rejoice, yea, and will rejoice" (Philippians 1:15-18 ASV).


Is there a difference between "bad people" (i.e., false teachers) and preachers who are perhaps correct doctrinally but are in need of an attitude adjustment? Yes, I think so. Nevertheless, we need to adopt Paul's perspective in this matter and rejoice because Christ is being preached. Competition, even among preachers, can be a good thing!  

Addendum:

Quite simply (but not to derogate your question), communicating the Word of God requires a clean vessel. Yes, even a person who is harboring and hiding a besetting sin can still preach and teach, but the hypocrisy of attempting to preach the truth while living a lie will eventually catch up with him or her, and the result could be his or her exposure, and with exposure comes (for example) losing one's ministry. Even with repentance, the person may not be able to return to a fruitful ministry, partly because of personal factors, and/or partly from people's lack of trust in the messenger. 

Marshall McLuhan was definitely onto something when he said, "The medium IS the message." You cannot separate one from the other. In Christian terms, a sinful messenger is lacking in spiritual power, spiritual insight, and spiritual sensitivity. Perhaps most serious of all is the sinner's failure to glorify God as much as he could have had he not lingered in unconfessed sin. 

While we all sin, and we "all stumble in many ways" (James 3:2), to glorify God we need to keep short accounts with God by confessing our sin, particularly before attempting to spread the word. Jesus condemned hypocrisy. A hypocritical spreader of the Word can do more harm than good, both to himself and to others, especially when his sin is "found out" (see Numbers 32:23).