Senate Democrats will need some Republican support to move legislation. They appear unlikely to eliminate the filibuster. And even if they loosen the rules that govern the budget-reconciliation process (which I think is likely) there will still be serious limits to what they can achieve through that process, which in any case can only be used rarely. That means they will need ten Republicans even for most bills that have the support of every Democrat, so Republicans have some significant leverage to make procedural demands at the outset of this Congress.
Equal numbers of Republicans and Democrats on committees and subcommittees seems a particularly likely arrangement, which may matter especially when it comes to nominations. Some of the limits of the 2001 arrangement, particularly the fact that it did not apply to House-Senate conference committees, have been made less relevant by changes in the culture of the Congress in the interim. There just aren’t many conference committees now, and the chambers tend to work out their differences by exchanging amendments instead. But there may have to be some creative new approach to the confirmation process for presidential appointees in a new arrangement, since things have changed a lot on that front in the Senate — most notably as the use of filibusters to hold up nominations has been much constrained.
A power-sharing agreement is not a sure thing. Some Senate Democrats may want to try to do without one, so they can flex more of their muscles as a majority. But at the end of the day, the Senate is tied, and without some formal arrangement the Democrats will find it very hard to run.