What's the count
 

Last weekend. Me and BU had same count while the field's scorekeeper as well as both opposing coaches had the same count which was different from ours. We relented. Was this correct? Thanks.

The umpires keep the count. The scorekeepers keep the outs.

Losing the count is never fun, and seldom ends well. You don't just lose the count, you also lose a little credibility.
Until you're secure in keeping the count (you never will be), your partner can help you if he keeps it as well. If I'm doing the bases I try to keep it, and I'll ask my partner in pregame to "flash me the count regularly with runners on base." As the plate guy, it also helps to verbalize and/or flash the count regularly (I do it every pitch with runners on base).
You can use the official scorer as a backup in a pinch, but depending on what level you're working they might not even be keeping the count. Is a team charting pitches? The player keeping that chart is seldom quick enough to purposely give you the wrong count. Hey, there's always a video review :D!
Bob had the best suggestion - depend on your partner.

JJ

In his case I'd bet on the scorekeeper plus two managers. :)

When I was pretty new at this, I lost track of the count quite often. As JJ said, you lose credibility.
For me, most of my problem was due to being overly dependent on my ball/strike indicator. Since then I try to not even look at my clicker, although I still work with one.
Verbalizing it frequently helps a lot. Any time I am 2 pitches or more into the count, I loudly verbalize before every pitch and incorporate the strike count into my call (ie. TWO!!!)
Also, when you are BU, keep count as well and have previously determined hand signals which HP can request help from you without the spectators knowing the count was lost.